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Jan 23, 2000, 3:00:00 AM1/23/00

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Recently in sci.math a question having to do with discontinuities

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.

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

Mar 16, 2015, 2:50:03 PM3/16/15

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воскресенье, 23 января 2000 г., 10:00:00 UTC+2 пользователь Dave L. Renfro написал:

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