HISTORICAL ESSAY ON CONTINUITY OF DERIVATIVES
Dave L. Renfro
of derivatives was asked. This issue seems to arise often in
math newsgroups. Since I didn't feel like grading the two
stacks of quizzes I have next to me today (it's Saturday--the
quizzes can wait until tomorrow), I decided to write a short
historical survey on the continuity of derivatives.
I. INTRODUCTION.
As Zdislav V. Kovarik pointed out in
[sci.math 21 Jan 2000 02:50:02 -0500]
<http://forum.swarthmore.edu/epigone/sci.math/prolelwil>,
a derivative is a Baire one function, which implies that any
derivative is continuous on the complement of a first category set.
Moreover, since the set of points at which an arbitrary function
is not continuous is an F_sigma set, it follows that the
non-continuity points of any derivative must be an F_sigma
first category set. In addition, because every derivative
satisfies the intermediate value property, no derivative can
have a jump discontinuity. (This was also pointed out by Kovarik.)
The F_sigma first category result is sharp: Any F_sigma first
category subset of the reals is the set of non-continuity points
for some derivative. A construction for this can be found
in Chapter 3.2 (specifically, p. 34) of Bruckner's book [4].
If we want the non-continuity points of a derivative to be LARGE,
then there are two natural directions to proceed: We can ask that
the set have positive measure or we can ask that the set be
(topologically) dense. The set can even have a measure zero
complement (which is stronger than having positive measure
*and* being dense), since there exist F_sigma first category
sets whose complements have measure zero. Indeed, the set can
be so large from a measure standpoint that its complement could
have Hausdorff dimension zero (even Hausdorff h-measure zero for
any given admissible Hausdorff measure function h), for the
same reason. However, the "positive measure" and "dense"
classifications are where the historically interesting
developments lie.
II. DERIVATIVES DISCONTINUOUS ON A SET OF POSITIVE MEASURE.
In 1881 Volterra [28] published an example of a function having a
bounded derivative that was not continuous on a set of positive
measure. One of the more novel aspects of this construction
at that time was the use of a nowhere dense set having positive
measure. [Volterra had just published (same volume of the same
journal, in fact) an example of such a set. Independently,
H. Smith (1875) and Du Bois-Reymond (1880) had also obtained
such sets.] The effect of Volterra's example was significant.
It showed that Riemann's theory of integration, which even allowed
for the integration of functions having a dense set discontinuities,
wasn't strong enough to "undo" differentiation even in the
bounded derivative case. [Lebesgue integration suffices for
bounded derivatives, and the Denjoy, Henstock/Kurzweil,
Khintchine, Perron, etc. integrals do better than this.]
For the details of the Volterra-type construction, see p. 33 of
Bruckner's book [4], Chapter 1.18 (pp. 54-56) of Bruckner^2/Thomson
[5], pp. 190-191 of Burrill/Knudsen [6], pp. 56-57 of Hawkins [12],
pp. 490-491 of Hobson's Volume I [13], Example 6.2 (pp. 148-149) of
Jeffery [14], and Section 503 (pp. 501-502) of Pierpont [24].
III. DERIVATIVES DISCONTINUOUS ON A DENSE SET.
Du Bois-Reymond's paper [10], which contained the first
*published* example of a continuous nowhere differentiable
function [This is the example Weierstrass presented to the
Berlin Academy of Sciences on June 18, 1872. Unaware of
the Weierstrass example, Darboux presented examples at the
French Mathematical Society on March 19, 1873 and Jan. 28,
1874.], conjectures (falsely) on p. 32 that an everywhere
differentiable function cannot have proper maxima and minima in
every subinterval of its domain. Dini [9] thought Bois-Reymond's
conjecture was false (p. 383 of the German version), but was
unable to construct a counterexample. Note that any such function
has a derivative that is discontinuous on a dense set. Here's a
proof of this last statement ----------->>>>>>>>
Let f(x) be everywhere oscillating and differentiable on an
interval J. Then f' is not continuous at each point of
J(+) = {x in J: f'(x) > 0} and J(-) = {x in J: f'(x) < 0},
since each point in J(+) and J(-) can be approached by a sequence
x_n such that f'(x_n) = 0. [It is possible for f' to only be
continuous at the points where f'(x) = 0 (theorem 5 on p. 9 of
Marcus [21]), and it is possible for f' not to be continuous at some
points where f'(x) = 0 (theorem 2 on p. 4 of Bruckner [2] gives
an example where the difference is at least one point; Cater [7]
gives an example such that f' is not continuous on a set whose
complement has measure zero and yet f' = 0 on a set of positive
measure in each interval of its domain).] Now observe that both
J(+) and J(-) are dense in J. [If a < b are in J(+), say, then
f'(c) = 0 for some c in [a,b]. Hence, f' maps [a,b] onto the
interval [0, max {f'(a), f'(b)}], since derivatives (whether
continuous or not) have the intermediate value property.]
Hankel [11] made an attempt at such a construction (pp. 81-84),
but was not able to obtain one. [I do not know whether Hankel
simply points out that he was unable to construct such a function
or whether he made an attempt that was later shown to be invalid.]
Kopcke attempted several constructions in the late 1880's
(papers which appeared in the journal Math. Ann. and the journal
Mitteil. Math. Gesell.), finally obtaining a satisfactory example
and proof around 1890 [17]. A simplification of Kopcke's example
was given by Pereno in 1897 [22]. Pereno's construction can be
found on pp. 412-421 of volume 2 of Hobson's book [13]. Schoenflies
[27] gave a general account of such functions in 1901. Nonetheless,
Denjoy [8] questioned the rigor, or at least the clarity, of these
early arguments and gave several constructions himself in 1915.
In 1927 Zalcwasser [32] proved the following: Given arbitrary
disjoint countable subsets A and B of the reals, there exists a
function having a bounded derivative such that A is the set of its
strict local maxima and B is the set of its strict local minima.
By choosing each of A and B to be dense, we get an everywhere
oscillating differentiable function. A much simpler
construction of Zalcwasser's result was given by Kelar in
1980 [16]. (See also Cater [7].)
In 1976 Cliff Weil [29] published an elegant Baire category
argument for the existence of everywhere differentiable and nowhere
monotone functions. Let D be the collection bounded derivatives g
(i.e. functions g: R --> R such that g is bounded and there exists
f such that f'(x) = g(x) for all x in R) such that g is zero
on a dense set, and put the sup metric on D. Then the set of
functions in D that are positive on one dense set and negative
on another dense set is the complement of a first category set
in D. [The proof can also be found on pp. 24-25 of Bruckner's
book [4], but note that Bruckner's definition of the space I called
D inadvertently omits the zero function, and therefore it is not
complete.]
Shortly after this Weil [30] published a proof that in the space
of bounded derivatives with the sup norm, all but a first category
set of such functions are discontinuous almost everywhere (in
the sense of Lebesgue measure).
Putting the first observation made in the Introduction next to
this last result makes for an interesting comparison:
(A) Every derivative is continuous at the Baire-typical point.
(B) The Baire-typical derivative is not continuous at the
Lebesgue-typical point.
IV. PROOF THAT ANY EVERYWHERE OSCILLATING DIFFERENTIABLE
FUNCTION FAILS TO BE RIEMANN INTEGRABLE ON EVERY SUBINTERVAL.
If f is everywhere oscillating and differentiable on an interval
J, then f' fails to be Riemann integrable on every subinterval of
J. [Note it suffices to prove that f' fails to be Riemann integrable
on J, since f will also be everywhere oscillating and differentiable
on every subinterval of J.] I don't know when this was first
observed (perhaps Schoenflies [27] in 1901?), but a proof appears
on p. 354 of the first edition (1907) of Hobson's book. There is an
easy proof of this fact due to B. K. Lahiri [19] that makes use
of the Denjoy-Clarkson property of derivatives. Recall that every
derivative f' has the intermediate value property. This implies
that each set E(a,b) = {x in J: a < f'(x) < b} is either empty
or has cardinality of the continuum. Denjoy (1916) (later
rediscovered by J. A. Clarkson in 1947) improved this by showing
that each of the sets E(a,b) is either empty or has positive
measure. [Further strengthenings of this result have been made
by Zahorski and Cliff Weil.] Let J(+) = {x in J: f'(x) > 0} and
recall from above that f' is not continuous at each point of
J(+). Choose a < b in the image of J(+) under f'. Then E(a,b)
is contained in J(+) and E(a,b) has positive measure by the
Denjoy-Clarkson property. Therefore, J(+), and hence also the
set of points at which f' is not continuous, has positive
measure, which prevents f' from being Riemann integrable on J.
V. REFERENCES.
[1] J. Blazek, E. Borak, and Jan Maly, "On Kopcke and Pompeiu
functions", Casopis pro Pest. Mat. 103 (1978), 53-61.
[2] Andrew M. Bruckner, "On derivatives with a dense set of
zeros", Rev. Roum. Math. Pures et Appl. 10 (1965), 149-153.
[3] Andrew M. Bruckner, "Some new simple proofs of old difficult
theorem", Real Analysis Exchange 9 (1983-84), 63-78. [Seven
proofs for the existence of a nowhere monotone function having
a bounded derivative are given by making use of the following
ideas: Baire category, density topology, extensions to
derivatives, products of derivatives, and changes of variable
and/or scale (3 proofs in this case).]
[4] Andrew M. Bruckner, DIFFERENTIATION OF REAL FUNCTIONS, 2'nd
edition, CRM Monograph Series 5, Amer. Math. Soc., 1994,
195 pages. [QA 304 .B78 1994] [The second edition is
essentially unchanged from the first edition (Lecture Notes
in Math. #659, Springer-Verlag, 1978), with the exception of
a new chapter on recent developments.]
[5] Andrew M. Bruckner, Judith B. Bruckner, and Brian S. Thomson,
REAL ANALYSIS, Prentice-Hall, 1997, 713 pages.
[QA 300 .B74 1997]
[6] Claude W. Burrill and John R. Knudsen, REAL VARIABLES, Holt,
Rinehart and Winston, 1969, 419 pages. [A fairly elementary
presentation of a derivative that is not continuous at each
point of the Cantor set is given on pp. 191-192. (Virtually
no modifications are needed to obtain the same result for
any closed nowhere dense set replacing the Cantor set.)]
[7] F. S. Cater, "Functions with preassigned local maximum points",
Rocky Mountain J. Math. 15 (1985), 215-217.
[8] Arnaud Denjoy, "Sur les fonctions derivees sommables", Bull. Soc.
Math. France 43 (1915), 161-248. [Four examples of everywhere
oscillating differentiable functions are constructed on
pp. 211-237.]
[9] Ulisse Dini, FONDAMENTI PER LA TEORICA DELLA FUNZIONI DI
VARIABILI REALI, Pisa, 1878. [The better known 554 page German
translation, in corporation with Jacob Luroth and Adolf Schepp,
appeared in 1892.]
[10] Paul du Bois-Reymond, "Versuch einer classification der
willkurlichen functionen reeler argumente nach ihren
aenderungen in den kleinsten intervallen", Journal fur die
reine und angewandte Mathematik 79 (1875), 21-37.
[11] Herrmann Hankel, "Untersuchungen uber die unendlich oft
oscillirenden und unstetigen functionen", Math. Ann. 20
(1882), 63-112. [Publication of Hankel's 1870 Ph.D.
dissertation at the Univ. of Tubingen.]
[12] Thomas Hawkins, LEBESGUE'S THEORY OF INTEGRATION: ITS ORIGINS
AND DEVELOPMENT, Chelsea Publishing Company, 1975, 227 pages.
[QA 312 .H34 1975]
[13] Ernest W. Hobson, THE THEORY OF FUNCTIONS OF A REAL VARIABLE
AND THE THEORY OF FOURIER'S SERIES, Volumes I and II, Dover
Publications, 1957. [Reprint of 3'rd edition (1927) of
Volume I and of the 2'nd edition (1926) of Volume II.]
[14] R. L. Jeffery, THE THEORY OF FUNCTIONS OF A REAL VARIABLE,
Dover Publications, 1985, 232 pages. [QA 331.5 .J43 1985]
[Reprint of 1953 edition.]
[15] Yitzhak Katznelson and Karl Stromberg, "Everywhere
differentiable, nowhere monotone, functions", Amer. Math.
Monthly 81 (1974), 349-354. [The example constructed in
this paper can also be found as Example 13.2 (pp. 80-83)
in Rooij/Schikhof's book.]
[16] Vaclav Kelar, "On strict local extrema of differentiable
functions", Real Analysis Exchange 6 (1980-81), 242-244.
[17] Alfred Kopcke, "Uber eine durchaus differentiirbare, stetige
function mit oscillationen in jedem intervalle", Math. Ann.
32 (1889), 161-171. [See also "Nachtrag zu dem aufsatze
'Uber eine durchaus ...", Math. Ann. 35 (1890), 104-109.]
[18] Thomas W. Korner, "A dense arcwise connected set of critical
points--molehills out of mountains", J. London Math. Soc.
(2) 38 (1988), 442-452. [Korner constructs a non-constant
everywhere differentiable function f: R^2 --> R with a bounded
derivative Df such that given any x, y in R^2 there exists a
continuous function h: [0,1] --> R^2 with h(0) = x, h(1) = y,
and such that Df evaluated at h(t) is 0 for each 0 < t < 1.]
[19] B. K. Lahiri, "A note on derivatives", Bull. Calcutta Math.
Soc. 50 (1958), 68-70.
[20] Jan S. Lipinski, "Sur les derivees de Pompeiu", Rev. Math.
Pures Appl. 10 (1965), 447-451.
[21] Solomon Marcus, "Sur les derivees dont les zeros forment un
ensemble frontiere partout dense", Rend. Circ. Mat. Palermo
(2) 12 (1963), 5-40. [An extensive survey, including a number
of new results, of functions f having a bounded derivative
such that both {x: f'(x) = 0} and its complement are dense.]
[22] Italo Pereno, "Sulle funzioni derivabili in ogni punto ed
infinitamente oscillanti in ogni intervallo", Giornale di
Matematiche 35 (1897), 132-149. [Pereno's construction of a
differentiable function that is everywhere oscillating can be
found on pp. 412-421 of volume 2 of Hobson's book.]
[23] Bruce Peterson, "A function with a discontinuous derivative",
Amer. Math. Monthly 89 (1982), 249-250, 263. [A differentiable
function f on the reals such that f' is not continuous at x=0,
f'(x) > 0 if x isn't 0, and f'(0) = 0. Note that the continuous
extension of (x^2)*sin(1/x) takes on all values in [-1,1] in
every open interval containing x=0.]
[24] James Pierpont, THE THEORY OF FUNCTIONS OF REAL VARIABLES,
Volume II, Ginn and Company, 1912, 645 pages. [See Example 6.2
(pp. 148-149) for the Volterra example and Sections 538-539:
Pompeiu Curves (pp. 542-546).]
[25] Dimitrie Pompeiu, "Sur les fonctions derivees", Math. Ann. 63
(1907), 326-332. [A construction of a function f having a
bounded derivative such that both the set where f' = 0 and
the set where f' \= 0 are dense. (This is not as strong a
requirement as having f' > 0 on a dense set and f' < 0 on
a dense set. Indeed, it is not difficult to construct a
function f such that f' >= 0 everywhere, with f' = 0 on a
dense set and f' > 0 on a dense set.) Note that f' is not
continuous on a dense set and f' is not Riemann integrable
in any subinterval of its domain (same proof as in part
IV above). For constructions of Pompeiu functions, see
Example 5.2 (pp. 205-206) of Bruckner^2/Thomson's book,
Section 538 (pp. 542-543) of Pierpont's book, and Example
13.3 (pp. 83-84) of Rooij/Schikhof's book.]
[26] A. C. M. Van Rooij and W. H. Schikhof, A SECOND COURSE ON REAL
FUNCTIONS, Cambridge University Press, 1982, 200 pages.
[QA 331 (I don't know the complete call number.)]
[27] Author Schoenflies, "Ueber die oscillirenden differenzirbaren
functionen", Math. Ann. 54 (1901), 553-563.
[28] Vito Volterra, "Sui principii del calcolo integrale", Giornale
di Matematiche 19 (1881), 333-372. [Besides the construction
of a derivative that fails to be continuous on a set of
positive measure, Volterra utilizes the modern notion "lim sup
of a function at a point". I don't know to what extent this
notion had been (correctly) defined prior to this.]
[29] Clifford Weil, "On nowhere monotone functions", Proc. Amer.
Math. Soc. 56 (1976), 388-389.
[30] Clifford Weil, "The space of bounded derivatives", Real
Analysis Exchange 3 (1977-78), 38-41.
[31] Zygmunt Zahorski, "Sur la primiere derivee", Trans. Amer. Math.
Soc. 69 (1950), 1-54. [At the end of Section 4, Zahorski gives
17 references to constructions of differentiable functions that
are everywhere oscillating (following a construction of his
own in this paper), 10 of which do not appear in my present
list. (These are Zahorski's # 1, 2, 3, 12, 14, 19, 20, 21,
28, and 33.)]
[32] Z. Zalcwasser, "On Kopcke functions" (Polish), Prace Mat. Fiz.
35 (1927-28), 57-99.
Dave L. Renfro
