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# References for Continuity Sets

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### Dave L. Renfro

Nov 25, 2006, 6:00:25â€¯PM11/25/06
to
This post is intended to archive some literature references
for characterizations of the continuity sets of monotone,
arbitrary, Baire one, and Riemann integrable functions. I'll
probably add to and/or revise this list as I become aware of
additional references. At present, it only includes references
I have on hand and was able to (today) locate these results in.
Although I know of several references that ought be utilized
(Hewitt/Stromberg, Natanson, etc.), I decided to go ahead and
post this now, and not wait until I get a chance to look
though the resources of a university library.

In what follows, "countable" means finite or countably infinite,
and R denotes the set of real numbers with their usual metric
and ordering properties.

We say a real-valued function f is Baire one if f is the
pointwise limit of some sequence of continuous functions.
Examples of Baire one functions are functions with countably
many points of continuity, semi-continuous functions,
derivatives, and functions f:R^2 --> R that are separately
continuous in each variable (but not necessarily functions
from R^n to R for n > 2).

Given a function f:R --> R, we let C(f) and D(f) denote the sets
of continuity and discontinuity points, respectively, of f.

--------------------------------------------------------------

THEOREM 1: If f:R --> R is monotone (or even of
bounded variation), then D(f) is countable.

THEOREM 1': If E is any countable subset of R, then
there exists a strictly increasing function
f:R --> R such that D(f) = E.

THEOREM 2: If f:R --> R is an arbitrary function, then
D(f) is an F_sigma subset of R.

THEOREM 2': Given any F_sigma subset E of R, then there
exists a function f:R --> R such that D(f) = E.

THEOREM 3: If f:R --> R is a Baire one function, then D(f)
is an F_sigma meager (= first category) subset of R.

REMARK: This implies that, for each Baire one function,
C(f) is dense in R, c-dense in R, and even co-meager
in every open interval.

THEOREM 3': Given any F_sigma meager subset E of R, then
there exists a Baire one function such that
D(f) = E. In fact, f can be chosen to be
semi-continuous or to be a bounded derivative.

REMARK: A meager F_sigma set can be c-dense in R, have a
measure zero complement, or to even have a Haudsorff
h-measure zero complement for any pre-assigned measure
function h.

THEOREM 4: If f:[a,b] --> R is Riemann integrable, then
D(f) is an F_sigma meager & measure zero subset
of R.

THEOREM 4': Given any F_sigma meager & measure zero subset E
of [a,b], then there exists a Riemann integrable
function f:[a,b] --> R such that D(f) = E.

--------------------------------------------------------------

PROOFS OF 1:

Bo [3] (Section 22, bottom of p. 159)
BBT [6] (Exercise 1:3.14 gives a method to be verified)
BBT [7] (Theorem 5.60, p. 247)
B/K [9] (Chapter 13-1, Corollary 2, p. 275)
C [10] (Chapter 2, Theorem 2.17, p. 32)
F [13] (Chapter 2.1, bottom of p. 79)
G/O [15] (Chapter 2, Exercise 2.1.1.14, p. 48)
Gor [19] (Solution to Exercise 5.4, p. 296)
K/K [25] (Chapter 1, Theorem 1.1.3, pp. 20-21)
O [31] (Theorem 7.8, p. 35)
R/S [33] (Theorem 1.2, p. 12)
Tay [34] (Chapter 9-1, Theorem 9-1 I, p. 380)
Tor [35] (Chapter 3.4, Exercise 4.4, p. 41; solution on p. 373)
W/Z [37] (Chapter 2, Theorem 2.8, p. 19)

PROOFS OF 1':

A [1] (Chapter 4.7, outlined on p. 128)
Be [2] (Chapter 8, Example 8.1.4, pp. 118-119)
Bo [3] (Section 22, pp. 159-160)
BBT [6] (Exercise 1:3.15 gives a construction to be verified)
BBT [7] (end of Section 5.9.2, p. 248)
B/K [9] (Chapter 13-1, Example 13-1, pp. 275-276)
C [10] (end of Chapter 2 material and Exercise 34, pp. 32-33)
G/O [14] (Chapter 2, Example 18, p. 28)
G/O [15] (Chapter 2, Exercise 2.1.1.14, pp. 48-49)
J [23] (Chapter 5, Exercise 5.7, p. 138)
K/K [25] (Chapter 1, Exercises 2 & 8, pp. 34 & 35
O [31] (Theorem 7.8, p. 35)
P [32] (Section 467, pp. 462-463)
R/S [33] (Theorem 1.2 & Corollary 1.3, p. 12)
Tor [35] (Chapter 3.4, Exercise 4.4, p. 41; solution on p. 373)

HISTORY: I believe Ludwig Scheeffer (1884) was the first
to obtain Theorem 1'. See pp. 74-75 of Hawkins'
"Lebesgue's Theory of Integration".

For a detailed discussion of Dini derivates of monotone
functions, see the following post.

ESSAY ON NON-DIFFERENTIABILITY POINTS OF MONOTONE FUNCTIONS

PROOFS OF 2:

A [1] (Chapter 4.6, outlined on pp. 126-127)
BBT [7] (Chapter 6.7.2, Theorem 6.28, p. 277)
C [10] (Chapter 9, Theorem 9.2, p. 129)
C/S [12] (Theorem 3.3)
F [13] (Chapter 2.1, Theorem 2.6, pp. 82-84)
Gof [16] (Chapter 7, Section 6, Theorem 5', p. 89)
Gol [17] (Chapter 5.6, Theorem 5.6E, p. 144)
G/L/M/P [18] (Chapter 3, Problem 1.7, p. 26; solution on p. 168)
K/N [24] (Problem 1.7.14, p. 33; Solution on p. 202)
O [31] (Theorem 7.1, p. 31)
R/S [33] (Section 7, Theorem 7.5, p. 44)

PROOFS OF 2':

BBT [7] (Section 6.7.2, Theorem 6.28, pp. 277-279)
F [13] (Chapter 2.1, Theorem 2.6, pp. 82-84)
G/O [14] (Chapter 2, Example 23, pp. 30-31)
G/O [15] (Chapter 2, Exercise 2.1.1.12, pp. 48-49)
G/L/M/P [18] (Chapter 3, Problem 1.8b, p. 26; solution on p. 168)
Ha [20] (Chapter 3, Section 3, Satz V, pp. 201-202)
Ha [21] (Chapter 3, Section 26, Theorem 26.4.3, pp. 193-194)
Ho [22] (Section 237, pp. 314-316)
K/N [24] (Problem 1.7.16, p. 34; Solution on p. 203)
O [31] (Theorem 7.2, pp. 31-32)
P [32] (Section 473, pp. 467-468)
R/S [33] (Section 7, Exercises 7.G & 7.H, p. 45 give outline)

Kim [27] (see also C/S [12]) proves Theorem 2' for metric spaces
having no isolated points (more generally, for F_sigma sets not
containing any points isolated in the metric space) and functions
f:X --> R. [They were apparently unaware that the result in Hahn's
1932 book [21] (pp. 193-194) is also for metric spaces.] Bolstein
[4] proves a generalization of Hahn and Kim's result for a class
of topological spaces that includes first countable spaces,
locally compact Hausdorff spaces, separable spaces, and
topological linear spaces. Chen/Su [11] prove that if X is
a topological space, then every F_sigma subset of X is a
discontinuity set for some function f:X --> R if and only if
there exists an everywhere discontinuous real-valued function
on X.

HISTORY: Theorem 2' was first proved by William H. Young in
1903 [38] for functions f:R --> R [and apparently
independently by Lebesgue [30] (pp. 235-236) in 1904],
and generalized in 1905 [39] (pp. 376-377) to functions
f:R^n --> R. The proof that Young gave in 1903
(in German) is very similar to the exposition
(in English) that can be found on pp. 314-316 of
Hobson [22].

PROOFS OF 3:

Bo [3] (Section 18, pp. 123-126)
BBT [6] (Chapter 1.6, Theorem 1.19, pp. 22-23)
BBT [7] (Section 9.8, Theorem 9.39, pp. 422-423)
C [10] (Chapter 11, Theorem 11.20, p. 183)
F [13] (Chapter 4.1, Theorem 4.7, pp. 163-164)
Gor [19] (Theorem 5.16, pp. 77-78; Solution to Exercise 5.9, p. 298)
K/N [24] (Problem 1.7.20, p. 34; Solution on pp. 205-206)
Ke [26] (Section 24, Theorem 24.14, p. 193)
Ku [29] (Chapter 2, Section 31, Theorem 1, p. 394; pp. 397-398)
O [31] (Theorem 7.3, p. 32)
R/S [33] (Section 11, Theorem 11.4, pp. 67-68)
Tow [36] (Chapter 3, Section 28, Theorem 4, pp. 130-131)

REMARK: In the presence of Theorem 2, it suffices to show
that C(f) is dense in R.

HISTORY: This result was obtained independently by William
F. Osgood (1897) and RenÃ© Baire (1899).

PROOFS OF 3':

(semi-continuous result)
Gof [16] (Chapter 7, Exercise 7.5, p. 98 states the result)
Gor [19] (Solution to Exercise 5.18, pp. 302-303)

(bounded derivative result)
Br [5] (Chapter 3, Section 2, Theorem 2.1, p. 34)
B/L [8] (Theorem at bottom of p. 27)
Gof [16] (Chapter 9, Exercise 2.3, p. 120 states the result)
K/W [28]

HISTORY: Regarding the bounded derivative result, Bruckner
and Leonard [8] (bottom of p. 27) wrote the following
in 1966: "Although we imagine that this theorem is
known, we have been unable to find a reference."
I have found the result given in Exercise 2.3 on
p. 120 of a 1953 text by Casper Goffman, but nowhere
else prior to 1966 (including Goffman's Ph.D.
Dissertation).

PROOFS OF 4 & 4':

References omitted because this follows from Theorem 2
and the fact -- whose proof can be found in most any real
analysis text -- that a bounded function f:[a,b] --> R is
Riemann integrable if and only if D(f) has measure zero.

Regarding our more precise (and apparently stronger for one
direction) version, note that any F_sigma measure zero set
is a countable union of closed measure zero sets, and hence
a countable union of nowhere dense sets.

Interestingly, despite the ease in which this more precise
version follows from results in virtually every graduate
level real analysis text, I have not seen this more precise
version explicitly stated outside of a handful of research
papers. Rooij/Schikhof [33] comes the closest that I've seen.
Their Exercise 6.I (p. 42) asks the reader to verify that every
F_sigma measure zero set is meager and their Theorem 7.5 (p. 44)
implies that D(f) is F_sigma. Even Oxtoby [31], which gives
an extensive overview of various measure and category analogs,
doesn't mention that D(f) is meager for a Riemann integrable f,
despite having (1) a proof of the Riemann integrability
continuity condition (pp. 33-34), (2) a proof that any D(f)
set is F_sigma (p. 31), and (3) the observation that any
F_sigma Lebesgue measure set is meager (bottom of p. 51).

What makes this result more interesting is that for a Riemann
integrable function f, D(f) is actually "infinitely smaller
than" some meager-and-measure-zero sets. More precisely, there
exists a set E such that E is meager and E has measure zero
such that E cannot be covered by countably many F_sigma measure
zero sets (i.e. the discontinuity sets of Riemann integrable
functions). Thus, not only is it an understatement to describe
the size of the discontinuity sets of Riemann integrable
functions by saying they have measure zero (because they are
also small in the Baire category sense), but it's even an
understatement to describe their size by saying they are
simultaneously measure zero and meager! For more about the
size classification that discontinuity sets of Riemann
integrable functions belong to, see the following post.

HISTORICAL ESSAY ON F_SIGMA LEBESGUE NULL SETS

--------------------------------------------------------------

[1] Stephen Abbott, UNDERSTANDING ANALYSIS, Undergraduate Texts
in Mathematics, Springer-Verlag, 2001, xii + 257 pages.
[MR 2001m:26001; Zbl 966.26001]

[2] Alan F. Beardon, LIMITS: A NEW APPROACH TO REAL ANALYSIS,
Undergraduate Texts in Mathematics, Springer-Verlag, 1997,
x + 189 pages. [MR 98i:26001; Zbl 892.26003]

[3] Ralph P. Boas, A PRIMER OF REAL FUNCTIONS, 4'th edition
(revised and updated by Harold P. Boas), Carus Mathematical
Monographs #13R, Mathematical Association of America,
1996, xiv + 305 pages. [MR 97f:26001; Zbl 865.26001]

[4] Richard Bolstein, "Sets of points of discontinuity",
Proceedings of the American Mathematical Society 38 (1973),
193-197. [MR 47 #1014; Zbl 232.54014]

[5] Andrew M. Bruckner, DIFFERENTIATION OF REAL FUNCTIONS,
2'nd edition, CRM Monograph Series #5, American Mathematical
Society, 1994, xii + 195 pages. [The first edition was
published in 1978 as Springer-Verlag's Lecture Notes in
Mathematics #659. The second edition is essentially
unchanged from the first edition with the exception of a
new chapter on recent developments (23 pages) and 94
[MR 94m:26001; Zbl 796.26001]

[6] Andrew M. Bruckner, Judith B. Bruckner, and Brian S.
Thomson, REAL ANALYSIS, Prentice-Hall, 1997,
xiv + 713 pages. [Zbl 872.26001]

[7] Andrew M. Bruckner, Judith B. Bruckner, and Brian S.
Thomson, ELEMENTARY REAL ANALYSIS, Prentice-Hall, 2001,
xv + 677 pages. [There are 58 additional pages of
appendixes: (A) Background, (B) Hints for Selected
Exercises, (C) Subject Index.]

[8] Andrew M. Bruckner and John L. Leonard, "Derivatives",
American Mathematical Monthly 73 #4 (April 1966) [Part II:
Papers in Analysis, Herbert Ellsworth Slaught Memorial
Papers #11], 24-56. [MR 33 #5797; Zbl 138.27805]

[9] Claude W. Burrill and John R. Knudsen, REAL VARIABLES,
Holt, Rinehart and Winston, 1969, xii + 419 pages.
[MR 39 #4328; Zbl 201.38101]

[10] Neal L. Carothers, REAL ANALYSIS, Cambridge University
Press, 2000, xiv + 401 pages. [Zbl 997.26003]

[11] Wea-Chin Chen and Chen-Jyi Su, "The relation between
G_delta-set and continuity set", Tamkang Journal of
Mathematics 1 (1970), 103-108. [MR 43 #8057; Zbl 215.51701]

[12] Jiaming Chen and Sam Smith, "Cardinality of the set of
real functions with a given continuity set", to appear in
Pi Mu Epsilon Journal, approx. 7 pages.
http://www.sju.edu/~smith/Current_Courses/continuitysetsrev.pdf

[13] James Foran, FUNDAMENTALS OF REAL ANALYSIS, Monographs and
Textbooks in Pure and Applied Mathematics #144, Marcel
Dekker, 1991, xiv + 473 pages. [MR 94e:00002; Zbl 0744.26004]

[14] Bernard R. Gelbaum and John M. H. Olmsted, COUNTEREXAMPLES
IN ANALYSIS, Dover Publications, 1964/2003, xxiv + 195 pages.
[MR 30 204; Zbl 121.28902]

[15] Bernard R. Gelbaum and John M. H. Olmsted, THEOREMS AND
COUNTEREXAMPLES IN MATHEMATICS, Problem Books in Mathematics,
Springer-Verlag, 1990, xxxiv + 305 pages.

[16] Casper Goffman, REAL FUNCTIONS, Prindle, Weber & Schmidt,
1953/1967, x + 261 pages. [MR 14,855e; Zbl 53.22502]

[17] Richard R. Goldberg, METHODS OF REAL ANALYSIS, 2'nd edition,
John Wiley & Sons, 1976, x + 402 pages.
[MR 57 #12796; Zbl Zbl 348.26001]

[18] M. G. Goluzina, A. A. Lodkin, B. M. Makarov, and A. N.
Podkorytov, SELECTED PROBLEMS IN REAL ANALYSIS,
Translations of Mathematical Monographs #107, American
Mathematical Society, 1992, x + 370 pages.
[MR 93i:26001; Zbl 765.26001]

[19] Russell A. Gordon, THE INTEGRALS OF LEGESGUE, DENJOY,
PERRON, AND HENSTOCK, Graduate Studies in Mathematics #4,
American Mathematical Society, 1994, xii + 395 pages.
[MR 95m:26010; Zbl 807.26004]

[20] Hans Hahn, THEORIE DER REELLEN FUNKTIONEN [Theory of Real
Functions], Verlag von Julius Springer, Berlin, 1921,
vii + 600 pages. [JFM 48.0261.09]

[21] Hans Hahn, REELLE FUNKTIONEN [Real Functions], Akademie
Verlagsgesellschaft, Leipzig, 1932.
[Zbl 5.38903; JFM 58.0242.05]

[22] Ernest W. Hobson, THE THEORY OF FUNCTIONS OF A REAL VARIABLE
AND THE THEORY OF FOURIER'S SERIES, Volume I, Dover
Publications, 1927/1957, xvi + 732 pages.
[MR 19,1166a; Zbl 81.27702; JFM 53.0226.01]

[23] Ralph L. Jeffery, THE THEORY OF FUNCTIONS OF A REAL VARIABLE,
Dover Publications, 1953/1985, xiv + 232 pages.
[MR 13,216b and MR 86g:26001; Zbl 43.27901]

[24] Wieslawa J. Kaczor and Maria T. Nowak, PROBLEMS IN
MATHEMATICAL ANALYSIS II: CONTINUITY AND DIFFERENTIATION,
Student Mathematical Library #12, American Mathematical
Society, 2001, xiv + 398 pages.
[MR 2002c:26001; Zbl 969.00006]

[25] Rangachary Kannan and Carole King Krueger, ADVANCED ANALYSIS
ON THE REAL LINE, Universitext, Springer-Verlag, 1996,
x + 259 pages.

[26] Alexander S. Kechris, CLASSICAL DESCRIPTIVE SET THEORY,
Graduate Texts in Mathematics #156, Springer-Verlag,
1995, xviii + 402 pages. [MR 96e:03057; Zbl 819.04002]

[27] Sung Soo Kim, "A characterization of the set of points of
continuity of a real function", American Mathematical Monthly
106 (1999), 258-259.

[28] John Clayton Klippert and Geoffrey Williams, "On the
existence of a derivative continuous on a G_delta",
International Journal of Mathematical Education in
Science and Technology 35 (2004), 91-99.

[29] Kazimierz Kuratowski, TOPOLOGY, Volume I, Academic Press,
1966, xx + 560 pages. [MR 36 #840; Zbl 158.40802]

de classe 1" [On characteristic properties for functions
32 (1904), 229-242. [JFM 35.0420.01]
http://www.numdam.org/numdam-bin/browse?j=BSMF

[31] John C. Oxtoby, MEASURE AND CATEGORY, 2'nd edition,
Graduate Texts in Mathematics #2, Springer-Verlag,
1980, x + 106 pages. [MR 81j:28003; Zbl 435.28011]

[32] James Pierpont, THE THEORY OF FUNCTIONS OF REAL VARIABLES,
Volume II, Ginn and Company, 1912, xiv + 645 pages.
[JFM 43.0476.01]

[33] Arnoud C. M. Van Rooij and Wilhelmus H. Schikhof, A SECOND
COURSE ON REAL FUNCTIONS, Cambridge University Press, 1982,
xiii + 200 pages. [MR 83j:26001; Zbl 474.26001]

[34] Angus E. Taylor, GENERAL THEORY OF FUNCTIONS AND INTEGRATION,
Dover Publications, 1965/1985, x + 437 pages.
[MR 31 #2358; Zbl 135.11301]

[35] Alberto Torchinsky, REAL VARIABLES, Addison-Wesley, 1988,
xii + 403 pages. [MR 89d:00003; Zbl 649.26002]

[36] Edgar Terome Townsend, FUNCTIONS OF REAL VARIABLES, Henry
Holt and Company, 1928, x + 405 pages. [JFM 54.0268.07]

[37] Richard L. Wheeden and Antoni Zygmund, MEASURE AND INTEGRAL:
AN INTRODUCTION TO REAL ANALYSIS, Marcel Dekker, 1977,
x + 274 pages. [MR 58 #11295; Zbl 362.26004]

[38] William H. Young, "Uber die Einteilung der unstetigen
Funktionen und die Verteilung ihrer Stetigkeitspunkte"
[On the structure of discontinuous functions and the
distribution of their points of continuity],
Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften
in Wien [Wiener Bericht] 112(2a) (1903), 1307-1317.
[JFM 34.0411.03]

[39] William H. Young, "Ordinary inner limiting sets in the plane
or higher space}, Proceedings of the London Mathematical
Society (2) 3 (1905), 371-380.

--------------------------------------------------------------

Dave L. Renfro

### Dave L. Renfro

Dec 9, 2006, 5:44:32â€¯PM12/9/06
to
This is a slightly expanded and corrected version of a post
I made two weeks ago, on November 25.

This post is intended to archive some literature references
for characterizations of the continuity sets of monotone,
arbitrary, Baire one, and Riemann integrable functions.

I did not include a reference for Theorem 1 unless I also
cite that reference for at least one of the other theorems.

--------------------------------------------------------------

--------------------------------------------------------------

PROOFS OF 1:

Boa [4] (Section 22, bottom of p. 159)
BBT [7] (Exercise 1:3.14 gives a method to be verified)
BBT [8] (Theorem 5.60, p. 247)
B/K [10] (Chapter 13-1, Corollary 2, p. 275)
Car [11] (Chapter 2, Theorem 2.17, p. 32)
For [15] (Chapter 2.1, bottom of p. 79)
G/O [17] (Chapter 2, Exercise 2.1.1.14, p. 48)
Gor [21] (Solution to Exercise 5.4, p. 296)
Hob [24] (Sections 227 & 239, pp. 304 & 318)
K/K [27] (Chapter 1, Theorem 1.1.3, pp. 20-21)
Kha [29] (Chapter 2, Theorem 1, p. 56 + middle p. 57)
Nag [36] (Chapter 2, Section 2.4.1, pp. 90-91)
Oxt [37] (Theorem 7.8, p. 35)
Ran [40] (Chapter 6, Section 8, Theorem 1, pp. 337-338)
R/S [41] (Theorem 1.2, p. 12)
Spr [42] (Section 26, Corollary 26.6, p. 175)
Tor [43] (Chapter 3.4, Exercise 4.4, p. 41; solution on p. 373)

PROOFS OF 1':

Abb [1] (Chapter 4.7, outlined on p. 128)
Bea [2] (Chapter 8, Example 8.1.4, pp. 118-119)
Boa [4] (Section 22, pp. 159-160)
BBT [7] (Exercise 1:3.15 gives a construction to be verified)
BBT [8] (end of Section 5.9.2, p. 248)
B/K [10] (Chapter 13-1, Example 13-1, pp. 275-276)
Car [11] (end of Chapter 2 material and Exercise 34, pp. 32-33)
Cha [12] (Chapter 5, Section 3, Exercise C, p. 206)
G/O [16] (Chapter 2, Example 18, p. 28)
G/O [17] (Chapter 2, Exercise 2.1.1.14, pp. 48-49)
Jef [25] (Chapter 5, Exercise 5.7, p. 138)
K/K [27] (Chapter 1, Exercises 2 & 8, pp. 34 & 35)
Kha [29] (Chapter 2, Exercise 2, pp. 57-58)
Kul [32] (Chapter 4, Section 8, Exercise 4, p. 133)
Nag [36] (Chapter 2, Section 2.4.2, Theorem, pp. 92-93)
Oxb [37] (Theorem 7.8, p. 35)
Pie [38] (Section 467, pp. 462-463)
Ran [40] (Chapter 6, Section 8, Problem 1, p. 338)
R/S [41] (Theorem 1.2 & Corollary 1.3, p. 12)
Spr [42] (Section 26, Example 26.7, pp. 175-176)
Tor [43] (Chapter 3.4, Exercise 4.4, p. 41; solution on p. 373)

HISTORY: I believe Ludwig Scheeffer (1884) was the first
to obtain Theorem 1'. See pp. 74-75 of Hawkins'
"Lebesgue's Theory of Integration".

For a detailed discussion of Dini derivates of monotone
functions, see the following post.

ESSAY ON NON-DIFFERENTIABILITY POINTS OF MONOTONE FUNCTIONS

PROOFS OF 2:

Abb [1] (Chapter 4.6, outlined on pp. 126-127)
Ben [3] (Chapter 1.3.1, Proposition 1.17, p. 24)
BBT [8] (Chapter 6.7.2, Theorem 6.28, p. 277)
Car [11] (Chapter 9, Theorem 9.2, p. 129)
C/S [14] (Theorem 3.3)
For [15] (Chapter 2.1, Theorem 2.6, pp. 82-84)
Gof [18] (Chapter 7, Section 6, Theorem 5', p. 89)
Gol [19] (Chapter 5.6, Theorem 5.6E, p. 144)
G/L/M/P [20] (Chapter 3, Problem 1.7, p. 26; solution on p. 168)
K/N [26] (Problem 1.7.14, p. 33; Solution on p. 202)
Mit [35] (Exercise 3.11a, pp. 24-25)
Oxb [37] (Theorem 7.1, p. 31)
Ran [40] (Chapter 6, Section 8, Problem 2, p. 338)
R/S [41] (Section 7, Theorem 7.5, p. 44)

PROOFS OF 2':

BBT [8] (Section 6.7.2, Theorem 6.28, pp. 277-279)
For [15] (Chapter 2.1, Theorem 2.6, pp. 82-84)
G/O [16] (Chapter 2, Example 23, pp. 30-31)
G/O [17] (Chapter 2, Exercise 2.1.1.12, pp. 48-49)
G/L/M/P [20] (Chapter 3, Problem 1.8b, p. 26; solution on p. 168)
Hah [22] (Chapter 3, Section 3, Satz V, pp. 201-202)
Hah [23] (Chapter 3, Section 26, Theorem 26.4.3, pp. 193-194)
Hob [24] (Section 237, pp. 314-316)
K/N [26] (Problem 1.7.16, p. 34; Solution on p. 203)
Mit [35] (Exercise 3.11b, pp. 24-25)
Oxb [37] (Theorem 7.2, pp. 31-32)
Pie [38] (Section 473, pp. 467-468)
R/S [41] (Section 7, Exercises 7.G & 7.H, p. 45 gives outline)

Kim [30] (see also C/S [14]) proves Theorem 2' for metric spaces

having no isolated points (more generally, for F_sigma sets not
containing any points isolated in the metric space) and functions
f:X --> R. [They were apparently unaware that the result in Hahn's

1932 book [23] (pp. 193-194) is also for metric spaces.] Bolstein
[5] proves a generalization of Hahn and Kim's result for a class

of topological spaces that includes first countable spaces,
locally compact Hausdorff spaces, separable spaces, and

topological linear spaces. Chen/Su [13] prove that if X is

a topological space, then every F_sigma subset of X is a
discontinuity set for some function f:X --> R if and only if
there exists an everywhere discontinuous real-valued function

on X. Mitsis [35] proves Theorem 2' for any function f:X --> Y
such that X is a topological space, Y is a metric space, and
there exists a dense subset of X that has a dense complement.

HISTORY: Theorem 2' was first proved by William H. Young in

1903 [45] for functions f:R --> R [and apparently
independently by Lebesgue [34] (pp. 235-236) in 1904],
and generalized in 1905 [46] (pp. 376-377) to functions

f:R^n --> R. The proof that Young gave in 1903
(in German) is very similar to the exposition
(in English) that can be found on pp. 314-316 of

Hobson [24].

PROOFS OF 3:

Boa [4] (Section 18, pp. 123-126)
BBT [7] (Chapter 1.6, Theorem 1.19, pp. 22-23)
BBT [8] (Section 9.8, Theorem 9.39, pp. 422-423)
Car [11] (Chapter 11, Theorem 11.20, p. 183)
For [15] (Chapter 4.1, Theorem 4.7, pp. 163-164)
Gor [21] (Theorem 5.16, pp. 77-78; Solution to Exercise 5.9, p. 298)
K/N [26] (Problem 1.7.20, p. 34; Solution on pp. 205-206)
Kec [28] (Section 24, Theorem 24.14, p. 193)
Kur [33] (Chapter 2, Section 31, Theorem 1, p. 394; pp. 397-398)
Mit [35] (Exercise 3.13a, p. 25)
Oxt [37] (Theorem 7.3, p. 32)
Pug [39] (Chapter 3, Section 3, Exercises 18 & 22 & 23, pp. 189-190)
R/S [41] (Section 11, Theorem 11.4, pp. 67-68)
Tow [44] (Chapter 3, Section 28, Theorem 4, pp. 130-131)

REMARK: In the presence of Theorem 2, it suffices to show
that C(f) is dense in R.

HISTORY: This result was obtained independently by William
F. Osgood (1897) and RenÃ© Baire (1899).

PROOFS OF 3':

(semi-continuous result)
Gof [18] (Chapter 7, Exercise 7.5, p. 98 states the result)
Gor [21] (Solution to Exercise 5.18, pp. 302-303)

(bounded derivative result)
Ben [3] (Chapter 1.3.2, Proposition, 1.10, p. 30)
Bru [6] (Chapter 3, Section 2, Theorem 2.1, p. 34)
B/L [9] (Theorem at bottom of p. 27)
Gof [18] (Chapter 9, Exercise 2.3, p. 120 states the result)
K/W [31]

HISTORY: Regarding the bounded derivative result, Bruckner

and Leonard [9] (bottom of p. 27) wrote the following

in 1966: "Although we imagine that this theorem is
known, we have been unable to find a reference."
I have found the result given in Exercise 2.3 on
p. 120 of a 1953 text by Casper Goffman, but nowhere
else prior to 1966 (including Goffman's Ph.D.
Dissertation).

PROOFS OF 4 & 4':

References omitted because this follows from Theorem 2
and the fact -- whose proof can be found in most any real
analysis text -- that a bounded function f:[a,b] --> R is
Riemann integrable if and only if D(f) has measure zero.

Regarding our more precise (and apparently stronger for one
direction) version, note that any F_sigma measure zero set
is a countable union of closed measure zero sets, and hence
a countable union of nowhere dense sets.

Interestingly, despite the ease in which this more precise
version follows from results in virtually every graduate
level real analysis text, I have not seen this more precise
version explicitly stated outside of a handful of research

papers. Rooij/Schikhof [41] comes the closest that I've seen.

Their Exercise 6.I (p. 42) asks the reader to verify that every
F_sigma measure zero set is meager and their Theorem 7.5 (p. 44)

implies that D(f) is F_sigma. Even Oxtoby [37], which gives

an extensive overview of various measure and category analogs,
doesn't mention that D(f) is meager for a Riemann integrable f,
despite having (1) a proof of the Riemann integrability
continuity condition (pp. 33-34), (2) a proof that any D(f)
set is F_sigma (p. 31), and (3) the observation that any
F_sigma Lebesgue measure set is meager (bottom of p. 51).

What makes this result more interesting is that for a Riemann
integrable function f, D(f) is actually "infinitely smaller
than" some meager-and-measure-zero sets. More precisely, there
exists a set E such that E is meager and E has measure zero
such that E cannot be covered by countably many F_sigma measure

zero sets (the latter being the discontinuity sets of Riemann

integrable functions). Thus, not only is it an understatement
to describe the size of the discontinuity sets of Riemann
integrable functions by saying they have measure zero (because
they are also small in the Baire category sense), but it's even
an understatement to describe their size by saying they are
simultaneously measure zero and meager! For more about the size
classification that discontinuity sets of Riemann integrable
functions belong to, see the following post.

HISTORICAL ESSAY ON F_SIGMA LEBESGUE NULL SETS

--------------------------------------------------------------

[1] Stephen Abbott, UNDERSTANDING ANALYSIS, Undergraduate Texts
in Mathematics, Springer-Verlag, 2001, xii + 257 pages.
[MR 2001m:26001; Zbl 966.26001]

[2] Alan F. Beardon, LIMITS: A NEW APPROACH TO REAL ANALYSIS,
Undergraduate Texts in Mathematics, Springer-Verlag, 1997,
x + 189 pages. [MR 98i:26001; Zbl 892.26003]

[3] John J. Benedetto, REAL VARIABLE AND INTEGRATION WITH
HISTORICAL NOTES, Mathematische LeitfÃ¤den. Stuttgart:
B. G. Teubne, 1976, 278 pages. [MR 58 #28328; Zbl 336.26001]

Errata can be found at
http://www.math.umd.edu/~jjb/

[4] Ralph P. Boas, A PRIMER OF REAL FUNCTIONS, 4'th edition

(revised and updated by Harold P. Boas), Carus Mathematical
Monographs #13R, Mathematical Association of America,
1996, xiv + 305 pages. [MR 97f:26001; Zbl 865.26001]

[5] Richard Bolstein, "Sets of points of discontinuity",

Proceedings of the American Mathematical Society 38 (1973),
193-197. [MR 47 #1014; Zbl 232.54014]

[6] Andrew M. Bruckner, DIFFERENTIATION OF REAL FUNCTIONS,

2'nd edition, CRM Monograph Series #5, American Mathematical
Society, 1994, xii + 195 pages. [The first edition was
published in 1978 as Springer-Verlag's Lecture Notes in
Mathematics #659. The second edition is essentially
unchanged from the first edition with the exception of a
new chapter on recent developments (23 pages) and 94
[MR 94m:26001; Zbl 796.26001]

[7] Andrew M. Bruckner, Judith B. Bruckner, and Brian S.

Thomson, REAL ANALYSIS, Prentice-Hall, 1997,
xiv + 713 pages. [Zbl 872.26001]

[8] Andrew M. Bruckner, Judith B. Bruckner, and Brian S.

Thomson, ELEMENTARY REAL ANALYSIS, Prentice-Hall, 2001,
xv + 677 pages. [There are 58 additional pages of
appendixes: (A) Background, (B) Hints for Selected
Exercises, (C) Subject Index.]

[9] Andrew M. Bruckner and John L. Leonard, "Derivatives",

American Mathematical Monthly 73 #4 (April 1966) [Part II:
Papers in Analysis, Herbert Ellsworth Slaught Memorial
Papers #11], 24-56. [MR 33 #5797; Zbl 138.27805]

[10] Claude W. Burrill and John R. Knudsen, REAL VARIABLES,

Holt, Rinehart and Winston, 1969, xii + 419 pages.
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[12] Soo Bong Chae, LEBESGUE INTEGRATION, Monographs and
Textbooks in Pure and Applied Mathematics #58, Marcel
Dekker, 1980, x + 314 pages. [MR 83e:28001; Zbl 438.28001]

[13] Wea-Chin Chen and Chen-Jyi Su, "The relation between

G_delta-set and continuity set", Tamkang Journal of
Mathematics 1 (1970), 103-108. [MR 43 #8057; Zbl 215.51701]

[14] Jiaming Chen and Sam Smith, "Cardinality of the set of

real functions with a given continuity set", to appear in
Pi Mu Epsilon Journal, approx. 7 pages.
http://www.sju.edu/~smith/Current_Courses/continuitysetsrev.pdf

[15] James Foran, FUNDAMENTALS OF REAL ANALYSIS, Monographs and

Textbooks in Pure and Applied Mathematics #144, Marcel

Dekker, 1991, xiv + 473 pages. [MR 94e:00002; Zbl 744.26004]

[16] Bernard R. Gelbaum and John M. H. Olmsted, COUNTEREXAMPLES

IN ANALYSIS, Dover Publications, 1964/2003, xxiv + 195 pages.
[MR 30 204; Zbl 121.28902]

[17] Bernard R. Gelbaum and John M. H. Olmsted, THEOREMS AND

COUNTEREXAMPLES IN MATHEMATICS, Problem Books in Mathematics,
Springer-Verlag, 1990, xxxiv + 305 pages.

[MR 92b:00003; Zbl 702.00005]

[18] Casper Goffman, REAL FUNCTIONS, Prindle, Weber & Schmidt,

1953/1967, x + 261 pages. [MR 14,855e; Zbl 53.22502]

[19] Richard R. Goldberg, METHODS OF REAL ANALYSIS, 2'nd edition,

John Wiley & Sons, 1976, x + 402 pages.
[MR 57 #12796; Zbl Zbl 348.26001]

[20] M. G. Goluzina, A. A. Lodkin, B. M. Makarov, and A. N.

Podkorytov, SELECTED PROBLEMS IN REAL ANALYSIS,
Translations of Mathematical Monographs #107, American
Mathematical Society, 1992, x + 370 pages.
[MR 93i:26001; Zbl 765.26001]

[21] Russell A. Gordon, THE INTEGRALS OF LEGESGUE, DENJOY,

PERRON, AND HENSTOCK, Graduate Studies in Mathematics #4,
American Mathematical Society, 1994, xii + 395 pages.
[MR 95m:26010; Zbl 807.26004]

[22] Hans Hahn, THEORIE DER REELLEN FUNKTIONEN [Theory of Real

Functions], Verlag von Julius Springer, Berlin, 1921,
vii + 600 pages. [JFM 48.0261.09]

[23] Hans Hahn, REELLE FUNKTIONEN [Real Functions], Akademie

Verlagsgesellschaft, Leipzig, 1932.
[Zbl 5.38903; JFM 58.0242.05]

[24] Ernest W. Hobson, THE THEORY OF FUNCTIONS OF A REAL VARIABLE

AND THE THEORY OF FOURIER'S SERIES, Volume I, Dover
Publications, 1927/1957, xvi + 732 pages.
[MR 19,1166a; Zbl 81.27702; JFM 53.0226.01]

[25] Ralph L. Jeffery, THE THEORY OF FUNCTIONS OF A REAL VARIABLE,

Dover Publications, 1953/1985, xiv + 232 pages.
[MR 13,216b and MR 86g:26001; Zbl 43.27901]

[26] Wieslawa J. Kaczor and Maria T. Nowak, PROBLEMS IN

MATHEMATICAL ANALYSIS II: CONTINUITY AND DIFFERENTIATION,
Student Mathematical Library #12, American Mathematical
Society, 2001, xiv + 398 pages.
[MR 2002c:26001; Zbl 969.00006]

[27] Rangachary Kannan and Carole King Krueger, ADVANCED ANALYSIS

ON THE REAL LINE, Universitext, Springer-Verlag, 1996,

x + 259 pages. [MR 97m:26001; Zbl 855.26001

[28] Alexander S. Kechris, CLASSICAL DESCRIPTIVE SET THEORY,

Graduate Texts in Mathematics #156, Springer-Verlag,
1995, xviii + 402 pages. [MR 96e:03057; Zbl 819.04002]

[29] Alexander B. Kharazishvili, STRANGE FUNCTIONS IN REAL

ANALYSIS, Monographs and Textbooks in Pure and Applied

Mathematics #229, Marcel Dekker, 2000, viii + 297 pages.
[MR 2001h:26001; Zbl 942.26001]

[30] Sung Soo Kim, "A characterization of the set of points of

continuity of a real function", American Mathematical Monthly
106 (1999), 258-259.

[31] John Clayton Klippert and Geoffrey Williams, "On the

existence of a derivative continuous on a G_delta",
International Journal of Mathematical Education in
Science and Technology 35 (2004), 91-99.

[32] Robert Garlin Kuller, TOPICS IN MODERN ANALYSIS,
Prentice-Hall, 1969, viii + 296 pages.
[MR 38 #6328; Zbl 198.47602]

[33] Kazimierz Kuratowski, TOPOLOGY, Volume I, Academic Press,

1966, xx + 560 pages. [MR 36 #840; Zbl 158.40802]

de classe 1" [On characteristic properties for functions
32 (1904), 229-242. [JFM 35.0420.01]
http://www.numdam.org/numdam-bin/browse?j=BSMF

[35] Themis Mitsis, "Exercises in Classical Real Analysis",
not dated, 31 pages.
http://fourier.math.uoc.gr/~mitsis/notes/real.pdf

[36] BÃ©la SzÃ¶kefalvi-Nagy, INTRODUCTION TO REAL FUNCTIONS AND
ORTHOGONAL EXPANSIONS, University Texts in the Mathematical
Sciences, Oxford University Press, 1965, xii + 447 pages.
[MR 31 #5938; Zbl 128.05101]

[37] John C. Oxtoby, MEASURE AND CATEGORY, 2'nd edition,

Graduate Texts in Mathematics #2, Springer-Verlag,
1980, x + 106 pages. [MR 81j:28003; Zbl 435.28011]

[38] James Pierpont, THE THEORY OF FUNCTIONS OF REAL VARIABLES,
Volume II, 2'nd edition, Ginn and Company, 1912,

xiv + 645 pages. [JFM 43.0476.01]

[39] Charles Chapman Pugh, REAL MATHEMATICAL ANALYSIS,

2002, xii + 437 pages. [MR 2003e:00005; Zbl 1003.26001]

[40] John F. Randolph, BASIC REAL AND ABSTRACT ANALYSIS,
Academic Press, 1968, xii + 515 pages.
[MR 37 #1211; Zbl 165.37201]

[41] Arnoud C. M. Van Rooij and Wilhelmus H. Schikhof, A SECOND

COURSE ON REAL FUNCTIONS, Cambridge University Press, 1982,
xiii + 200 pages. [MR 83j:26001; Zbl 474.26001]

[42] David A. Sprecher, ELEMENTS OF REAL ANALYSIS, Dover
Publications, 1970/1987, viii + 343 pages.
[MR 89b:26001; Zbl 641.26001]

[43] Alberto Torchinsky, REAL VARIABLES, Addison-Wesley, 1988,

xii + 403 pages. [MR 89d:00003; Zbl 649.26002]

[44] Edgar Terome Townsend, FUNCTIONS OF REAL VARIABLES, Henry

Holt and Company, 1928, x + 405 pages. [JFM 54.0268.07]

[45] William H. Young, "Uber die Einteilung der unstetigen

Funktionen und die Verteilung ihrer Stetigkeitspunkte"
[On the structure of discontinuous functions and the
distribution of their points of continuity],
Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften
in Wien [Wiener Bericht] 112(2a) (1903), 1307-1317.
[JFM 34.0411.03]

[46] William H. Young, "Ordinary inner limiting sets in the plane

or higher space", Proceedings of the London Mathematical

Society (2) 3 (1905), 371-380. [JFM 36.0101.03]

--------------------------------------------------------------

Dave L. Renfro

### Dave L. Renfro

Dec 20, 2006, 4:23:27â€¯PM12/20/06
to
This is a slightly expanded version of the version I posted
on December 9. A more precise form of Theorem 2' is given
and new references are [24], [26], [36], [45], [46], and [48].

This post is intended to archive some literature references
for characterizations of the continuity sets of monotone,
arbitrary, Baire one, and Riemann integrable functions.

I did not include a reference for Theorem 1 unless I also
cite that reference for at least one of the other theorems.

In what follows, "countable" means finite or countably infinite,

and R denotes the set of real numbers with their usual metric
and ordering properties.

We say a real-valued function f is Baire one if f is the
pointwise limit of some sequence of continuous functions.
Examples of Baire one functions are functions with countably
many points of continuity, semi-continuous functions,
derivatives, and functions f:R^2 --> R that are separately
continuous in each variable (but not necessarily functions

from R^n to R for n > 2). A function is Baire two if it is
a pointwise limit of Baire one functions.

Given a function f:R --> R, we let C(f) and D(f) denote the sets
of continuity and discontinuity points, respectively, of f.

--------------------------------------------------------------

THEOREM 1: If f:R --> R is monotone (or even of
bounded variation), then D(f) is countable.

THEOREM 1': If E is any countable subset of R, then
there exists a strictly increasing function
f:R --> R such that D(f) = E.

THEOREM 2: If f:R --> R is an arbitrary function, then
D(f) is an F_sigma subset of R.

THEOREM 2': Given any F_sigma subset E of R, then there

exists a Baire two function f:R --> R such that
D(f) = E.

REMARK: Very few, if any, of the references point out that
f can be chosen to be a Baire two function, but the
actual constructions are clearly Baire two functions.

THEOREM 3: If f:R --> R is a Baire one function, then D(f)
is an F_sigma meager (= first category) subset of R.

REMARK: This implies that, for each Baire one function,
C(f) is dense in R, c-dense in R, and even co-meager
in every open interval.

THEOREM 3': Given any F_sigma meager subset E of R, then
there exists a Baire one function such that
D(f) = E. In fact, f can be chosen to be
semi-continuous or to be a bounded derivative.

REMARK: A meager F_sigma set can be c-dense in R, have a
measure zero complement, or to even have a Haudsorff
h-measure zero complement for any pre-assigned measure
function h.

THEOREM 4: If f:[a,b] --> R is Riemann integrable, then
D(f) is an F_sigma meager & measure zero subset
of R.

THEOREM 4': Given any F_sigma meager & measure zero subset E
of [a,b], then there exists a Riemann integrable
function f:[a,b] --> R such that D(f) = E.

--------------------------------------------------------------

PROOFS OF 1:

Boa [4] (Section 22, bottom of p. 159)
BBT [7] (Exercise 1:3.14 gives a method to be verified)
BBT [8] (Theorem 5.60, p. 247)
B/K [10] (Chapter 13-1, Corollary 2, p. 275)
Car [11] (Chapter 2, Theorem 2.17, p. 32)
For [15] (Chapter 2.1, bottom of p. 79)
G/O [17] (Chapter 2, Exercise 2.1.1.14, p. 48)
Gor [21] (Solution to Exercise 5.4, p. 296)
Hob [27] (Sections 227 & 239, pp. 304 & 318)
K/K [29] (Chapter 1, Theorem 1.1.3, pp. 20-21)
Kha [31] (Chapter 2, Theorem 1, p. 56 + middle p. 57)
Nag [39] (Chapter 2, Section 2.4.1, pp. 90-91)
Oxt [40] (Theorem 7.8, p. 35)
Ran [43] (Chapter 6, Section 8, Theorem 1, pp. 337-338)
R/S [44] (Theorem 1.2, p. 12)
Spr [47] (Section 26, Corollary 26.6, p. 175)
Tor [49] (Chapter 3.4, Exercise 4.4, p. 41; solution on p. 373)

PROOFS OF 1':

Abb [1] (Chapter 4.7, outlined on p. 128)
Bea [2] (Chapter 8, Example 8.1.4, pp. 118-119)
Boa [4] (Section 22, pp. 159-160)
BBT [7] (Exercise 1:3.15 gives a construction to be verified)
BBT [8] (end of Section 5.9.2, p. 248)
B/K [10] (Chapter 13-1, Example 13-1, pp. 275-276)
Car [11] (end of Chapter 2 material and Exercise 34, pp. 32-33)
Cha [12] (Chapter 5, Section 3, Exercise C, p. 206)

G/O [16] (Chapter 2, Example 18, p. 28)
G/O [17] (Chapter 2, Exercise 2.1.1.14, pp. 48-49)
Hau [24] (Chapter 9, Section 42.3, p. 285; for rationals)
Jef [27] (Chapter 5, Exercise 5.7, p. 138)
K/K [29] (Chapter 1, Exercises 2 & 8, pp. 34 & 35)
Kha [31] (Chapter 2, Exercise 2, pp. 57-58)
Kul [34] (Chapter 4, Section 8, Exercise 4, p. 133)
Nag [39] (Chapter 2, Section 2.4.2, Theorem, pp. 92-93)
Oxb [40] (Theorem 7.8, p. 35)
Pie [41] (Section 467, pp. 462-463)
Ran [43] (Chapter 6, Section 8, Problem 1, p. 338)
R/S [44] (Theorem 1.2 & Corollary 1.3, p. 12)
Spr [47] (Section 26, Example 26.7, pp. 175-176)
Tor [49] (Chapter 3.4, Exercise 4.4, p. 41; solution on p. 373)

HISTORY: I believe Ludwig Scheeffer (1884) was the first
to obtain Theorem 1'. See pp. 74-75 of Hawkins'
"Lebesgue's Theory of Integration".

For a detailed discussion of Dini derivates of monotone
functions, see the following post.

ESSAY ON NON-DIFFERENTIABILITY POINTS OF MONOTONE FUNCTIONS

PROOFS OF 2:

Abb [1] (Chapter 4.6, outlined on pp. 126-127)

Ben [3] (Chapter 1.3.1, Proposition 1.17, p. 24)
BBT [8] (Chapter 6.7.2, Theorem 6.28, p. 277)
Car [11] (Chapter 9, Theorem 9.2, p. 129)
C/S [14] (Theorem 3.3)

For [15] (Chapter 2.1, Theorem 2.6, pp. 82-84)
Gof [18] (Chapter 7, Section 6, Theorem 5', p. 89)
Gol [19] (Chapter 5.6, Theorem 5.6E, p. 144)
G/L/M/P [20] (Chapter 3, Problem 1.7, p. 26; solution on p. 168)
Hau [24] (Chapter 9, Section 42.3, pp. 284-285)
K/N [28] (Problem 1.7.14, p. 33; Solution on p. 202)
K/S [36] (Section 3.1, pp. 456-457)
Mit [38] (Exercise 3.11a, pp. 24-25)
Oxb [40] (Theorem 7.1, p. 31)
Ran [43] (Chapter 6, Section 8, Problem 2, p. 338)
R/S [44] (Section 7, Theorem 7.5, p. 44)
Sie [45] (Chapter 7, Section 4, Example 8, p. 214)
Sie [46] (Chapter 6, Section 72, pp. 183-185)
Sri [48] (Chapter 2, Section 2, 2.2.3 & 2.2.4, p. 54)

PROOFS OF 2':

BBT [8] (Section 6.7.2, Theorem 6.28, pp. 277-279)
For [15] (Chapter 2.1, Theorem 2.6, pp. 82-84)
G/O [16] (Chapter 2, Example 23, pp. 30-31)
G/O [17] (Chapter 2, Exercise 2.1.1.12, pp. 48-49)
G/L/M/P [20] (Chapter 3, Problem 1.8b, p. 26; solution on p. 168)
Hah [22] (Chapter 3, Section 3, Satz V, pp. 201-202)
Hah [23] (Chapter 3, Section 26, Theorem 26.4.3, pp. 193-194)
Hob [25] (Section 237, pp. 314-316)
K/N [28] (Problem 1.7.16, p. 34; Solution on p. 203)
Mit [38] (Exercise 3.11b, pp. 24-25)
Oxb [40] (Theorem 7.2, pp. 31-32)
Pie [41] (Section 473, pp. 467-468)
R/S [44] (Section 7, Exercises 7.G & 7.H, p. 45 gives outline)
Sie [46] (Chapter 6, Section 72, Example, p. 185)

Kim [32] (see also C/S [14]) proves Theorem 2' for metric spaces

having no isolated points (more generally, for F_sigma sets not
containing any points isolated in the metric space) and functions
f:X --> R. [They were apparently unaware that the result in Hahn's

1932 book [23] (pp. 193-194) is also for metric spaces.] Bolstein
[5] proves a generalization of Hahn and Kim's result for a class

of topological spaces that includes first countable spaces,
locally compact Hausdorff spaces, separable spaces, and

topological linear spaces. Chen/Su [13] prove that if X is

a topological space, then every F_sigma subset of X is a
discontinuity set for some function f:X --> R if and only if
there exists an everywhere discontinuous real-valued function

on X. Mitsis [38] proves Theorem 2' for any function f:X --> Y

such that X is a topological space, Y is a metric space, and
there exists a dense subset of X that has a dense complement.

HISTORY: Theorem 2' was first proved by William H. Young in
1903 [51] for functions f:R --> R [and apparently
independently by Lebesgue [37] (pp. 235-236) in 1904],
and generalized in 1905 [52] (pp. 376-377) to functions

f:R^n --> R. The proof that Young gave in 1903
(in German) is very similar to the exposition
(in English) that can be found on pp. 314-316 of

Hobson [25].

PROOFS OF 3:

Boa [4] (Section 18, pp. 123-126)
BBT [7] (Chapter 1.6, Theorem 1.19, pp. 22-23)
BBT [8] (Section 9.8, Theorem 9.39, pp. 422-423)
Car [11] (Chapter 11, Theorem 11.20, p. 183)
For [15] (Chapter 4.1, Theorem 4.7, pp. 163-164)
Gor [21] (Theorem 5.16, pp. 77-78; Solution to Exercise 5.9, p. 298)
Hau [24] (Chapter 9, Section 42.4, pp. 286-287)
Hob [25] (Section 231, pp. 309-310; for semi-continuous functions)
Hob [26] (Sections 185 & 190, pp. 264-265 & 273-274)
K/N [28] (Problem 1.7.20, p. 34; Solution on pp. 205-206)
Kec [30] (Section 24, Theorem 24.14, p. 193)
Kur [35] (Chapter 2, Section 31, Theorem 1, p. 394; pp. 397-398)
K/S [36] (Section 4.3, pp. 460-462)
Mit [38] (Exercise 3.13a, p. 25)
Oxt [40] (Theorem 7.3, p. 32)
Pug [42] (Chapter 3, Section 3, Exercises 18 & 22 & 23, pp. 189-190)
R/S [44] (Section 11, Theorem 11.4, pp. 67-68)
Tow [50] (Chapter 3, Section 28, Theorem 4, pp. 130-131)

REMARK: In the presence of Theorem 2, it suffices to show
that C(f) is dense in R.

HISTORY: This result was obtained independently by William
F. Osgood (1897) and RenÃ© Baire (1899).

PROOFS OF 3':

(semi-continuous result)
Gof [18] (Chapter 7, Exercise 7.5, p. 98 states the result)
Gor [21] (Solution to Exercise 5.18, pp. 302-303)

(bounded derivative result)

Ben [3] (Chapter 1.3.2, Proposition, 1.10, p. 30)

Bru [6] (Chapter 3, Section 2, Theorem 2.1, p. 34)
B/L [9] (Theorem at bottom of p. 27)
Gof [18] (Chapter 9, Exercise 2.3, p. 120 states the result)
K/W [33]

HISTORY: Regarding the bounded derivative result, Bruckner

and Leonard [9] (bottom of p. 27) wrote the following

in 1966: "Although we imagine that this theorem is
known, we have been unable to find a reference."
I have found the result given in Exercise 2.3 on
p. 120 of a 1953 text by Casper Goffman, but nowhere
else prior to 1966 (including Goffman's Ph.D.
Dissertation).

PROOFS OF 4 & 4':

References omitted because this follows from Theorem 2
and the fact -- whose proof can be found in most any real
analysis text -- that a bounded function f:[a,b] --> R is
Riemann integrable if and only if D(f) has measure zero.

Regarding our more precise (and apparently stronger for one
direction) version, note that any F_sigma measure zero set
is a countable union of closed measure zero sets, and hence
a countable union of nowhere dense sets.

Interestingly, despite the ease in which this more precise
version follows from results in virtually every graduate
level real analysis text, I have not seen this more precise
version explicitly stated outside of a handful of research

papers. Rooij/Schikhof [44] comes the closest that I've seen.

Their Exercise 6.I (p. 42) asks the reader to verify that every
F_sigma measure zero set is meager and their Theorem 7.5 (p. 44)

implies that D(f) is F_sigma. Even Oxtoby [40], which gives

an extensive overview of various measure and category analogs,
doesn't mention that D(f) is meager for a Riemann integrable f,
despite having (1) a proof of the Riemann integrability
continuity condition (pp. 33-34), (2) a proof that any D(f)
set is F_sigma (p. 31), and (3) the observation that any
F_sigma Lebesgue measure set is meager (bottom of p. 51).

What makes this result more interesting is that for a Riemann
integrable function f, D(f) is actually "infinitely smaller
than" some meager-and-measure-zero sets. More precisely, there
exists a set E such that E is meager and E has measure zero
such that E cannot be covered by countably many F_sigma measure

zero sets (the latter being the discontinuity sets of Riemann

integrable functions). Thus, not only is it an understatement
to describe the size of the discontinuity sets of Riemann
integrable functions by saying they have measure zero (because
they are also small in the Baire category sense), but it's even
an understatement to describe their size by saying they are
simultaneously measure zero and meager! For more about the size
classification that discontinuity sets of Riemann integrable
functions belong to, see the following post.

HISTORICAL ESSAY ON F_SIGMA LEBESGUE NULL SETS

--------------------------------------------------------------

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