The American Mathematical Monthly, Vol. 81, No. 4. (Apr., 1974), pp

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1 Everywhere Differentiable, Nowhere Monotone, Functions Y. Katznelson; Karl Stromberg The American Mathematical Monthly, Vol. 81, No. 4. (Apr., 1974), pp Stable URL: The American Mathematical Monthly is currently published by Mathematical Association of America. Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. The JSTOR Archive is a trusted digital repository providing for long-term preservation and access to leading academic journals and scholarly literature from around the world. The Archive is supported by libraries, scholarly societies, publishers, and foundations. It is an initiative of JSTOR, a not-for-profit organization with a mission to help the scholarly community take advantage of advances in technology. For more information regarding JSTOR, please contact support@jstor.org. Wed Jan 16 14:39:

2 19741 EVERYWHERE DIFFERENTIABLE NOWHERE MONOTONE FUNCTIONS 349 Again let 4:R3 -+ R x R3 be defined by taking the spatial section,of an inertial observer. Then q5*(d9-9) = dq5*9 - q5*9 = dq5*9- q5*y = (dive - a)dx A dy A dz vanishes, by hypothesis, for each inertial observer. By the same reasoning as before one concludes that d 9 -Y = 0, and we are finished. Work supported in part by N. S. F. grant G. P References 1. Y. Choquet-Bruhat, Geometric diffkrentielle et systkmes exterieurs, Dunod, Paris, Harley Flanders, Differential forms, Academic Press, New York, J. A. Wheeler, Gravitation as geometry-11, a chapter in the book Gravitation and Relativity, edited by H. Y. Chiu and W. F. Hoffmann, Benjamin, New York, E. T. Whittaker, A history of the theories of aether and electricity, Vol. 2, Harper and Brothers, New York, EVERYWHERE DIFFERENTIABLE, NOWHERE MONOTONE, FUNCTIONS Y. KATZNELSON AND KARL STROMBERG The purpose of this paper is to construct an example of a real-valued function on R that is everywhere diiferentiable but is monotone on no interval and to examine further peculiar properties of any such function. Examples of such functions are seldom given, or even mentioned, in books on real analysis. The first explicit construction of such a function was given by Kopcke (1889). An example due to Pereno (1897) is reproduced in [I], pp We believe that the present construction, which grew out of a question asked of the first named author by the second, is shorter, more elementary, and easier to understand than any that we have seen. We proceed through a sequence of easy lemmas. LEMMA1. Let r and s be real numbers. (i) If r > s > 0, then (r -s)/(r2-s2)< 2/r. (ii) If r>l and s>1, then (r+s-2)/(r2+s2-2)<2/s. Proof. Assertion (i) is obvious. Inequality (ii) is equivalent to But this too is obvious when r > 1 and s > 1.

3 350 Y. KATZNELSON AND KARL STROMBERG [April LEMMA2. Let $(x) = (1 + 1 x I)-* for x E R. Then l/(b-a) J:$(x)dx c 4min {$(a), $(b)) whenever a and b are distinct real numbers. Proof. We may obviously suppose that a < b. In case 0 2 a, we have by Lemma 1 that The case that b 2 0 follows from the evenness of 4.Thus we suppose that a < 0 < b. Then Lemma 1 yields LEMMA3. If $ is as in Lemma 2 and $ is any function of the form where c,,...,c, and A,,...,A, aye positive real numbers and a,,...,a, are any real numbers, then whenever a and b are distinct real numbers. Proof. This follows at once from Lemma 2 and the fact that LEMMA4. Let ($,):=, and each n define be any sequence of functions as in Lemma 3. For x E R Suppose that :=,$,(a) = s < co for some a E R. Then the series F(x) = Z,"=,Y,,(x) converges uniformly on every bounded subset of R, the function F is differentiable at a, and F'(a) = s. In particular, if C:==, $,,(t) = f(t) < co for all t E R, then F is difserentiable everywhere on R and F' = f.

4 19741 EVERYWHERE DIFFERENTIABLE NOWHERE MONOTONE FUNCTIONS 35 1 Proof. Let b E R satisfy b 5 1 a 1. Then, using Lemma 3, -b 5 x 5 b implies Thus, uniform convergence on [- b, b] follows from the Weierstrass M-test. To prove that F1(a) = s, let E > 0 be given. Choose N such that 10. E,"=,+,$,,(a) < E. Since each $, is continuous at a, there exists some 6 > 0 such that whenever 0 < I h I < S and 1 5 n $ N. Therefore, using Lemma 3 again, 0 < I h I < 6 implies that LEMMA5. Let Il;..,In be disjoint open intervals, let aj be the midpoint of Ij, and 'let E and y,,..., y,, be positive real numbers. Then there exists a as in Lemma 3 such that for each j Yj yj + E if XE Xi, (iii) $(x) c E if x 6 I1U '.-UIn. Proof. Choose cj = yj + el2 and write q5j(~)= cjq5(aj(x - aj)), where Aj is cho5en so large that q5j(x) < ~/2n if x 6 Ij (one needs only to check this inequality at an endpoint of Xi). Take $ = 4, ,. Since the Ij's are disjoint and since 4j takes its maximum value cj at aj, properties (i), (ii), and (iii) are evident. THEOREM. Let {aj}7=, and {pj}im_, be disjoint countable subsets of R. Then there exists a real-valued, everywhere differentiable, function F on R satisfying F1(aj) = 1, Ft(pj) < 1for all j, and 0 < Ft(x) 5 4. for all x.

5 352 Y. KATZNELSON AND KARL STROMBERG [April Proof. We obtain F as in Lemma 4 by first constructing F' = f = E,"=, $,, in such a way that more precisely, the partial sums f, = Z", 1 A,: fn(aj)> n (1 5j 5 n), IJ,, or, Supposing that this were done we would have and, choosing n >j, F1(aj)= lim f,(aj) = 1, n'm 0 < F1(x)= lim f,(x) 5 1, n+m m F1(Pj)= fn- t(pj> + $k(pj) k=n and thus we would have the desired F. We proceed inductively. First choose an open interval I with midpoint a, such that p, $I. Then apply Lemma 5 with E = yl = 4 to obtain f, = IJ, that satisfies A,, Bl, C,. Suppose that n > 1 and that f,-, and I),-,have been chosen which satisfy A,-,, B,-,, C,-,. Select disjoint open intervals I,,...,I, such that, for each j~ (1,...,n), isth them id point ofij, Ijn {PI,...,P,} = and fn-,(x) <f,-,(aj) + 6 for x E Ij, where Now apply Lemma 5, with E = 1/(2n. 2") and yj = 1 - (lln)-f,- to obtain IJ,. Plainly C, obtains. Also,(aj) (1 5 j j n),

6 19741 EVERYWHERE DIFFERENTIABLE NOWHERE MONOTONE FUNCTIONS 353 (1 5 j 5 n), and so A, obtains. To check B,, notice that if x E Ij, then while if x q! UYElIj, then This completes the proof. COROLLARY. Thereexists a real-valued, everywhere differentiable, function on R such that H is monotone on no subinterval of R and H' is bounded. Proof. Let {aj),?= and {Pj)j"=, be disjoint dense subsets of R. Apply the preceding theorem to obtain everywhere differentiable functions F and G on R such that H for all j and x. Now write H = F - G. Then for all j and x. Since {aj)jm,, and {Pi);= on an interval. are both dense, H cannot be monotone REMARKS. Let H be any function as in the preceding Corollary. (a) H has a local maximum and a local minimum in every interval of R. In fact, if a < b, we can choose a < p in [a, b] such that H1(a)> 0 and H1(P)< 0. Then H takes an absolute maximum value on [a,p] at some y E [a,pi. Clearly a < y < P, and so H(y) is a local maximum value for H. Similar reasoning produces a local minimum in [a, b]. (b) Since H' is bounded, it is clear from the mean value theorem that H is absolutely continuous on every closed interval of R. Thus H' is Lebesgue integrable on each such interval and for all x E R.

7 354 JUDITH V. GRABINER [April (c) H' is not Riemann integrable on any closed interval [a, b], for assume that it is. Then H' is continuous a.e. on [a, b]. But it is clear that H'(t) = 0 if H' is continuous at t,and so H' = 0 a.e. on [a, b]. It follows from (b) that H is a constant on [a, b] -a palpable contradiction. (d) H' is of Baire class one, being the pointwise limit of the continuous functions H,(x) = n[h(x + lln) - H(x)],and so the set of points at which H' is continuous is residual; i.e., its complement is of first category. (e) WriteA = {x: Hf(x) >O)andB = {x: H1(x) <O). Thus AnI and B n I both have positive Lebesgue measure for every interval I. In fact, assuming that there exists some interval I = [a, b] such that Bn I has measure zero, it follows that H' 2 0 a.e. on I. Therefore, since for all x E I, we conclude that H is nondecreasing on I- a contradiction. Similarly, if A n I had measure zero, then H would be nonincreasing on I. Reference 1. E. W. Hobson, Theory of Functions of a Real Variable 11, Dover, New York, IS MATHEMATICAL TRUTH TIME-DEPENDENT? JUDITH V. GRABINER 1. Introduction. Is mathematical truth time-dependent? Our immediate impulse is to answer no. To be sure, we acknowledge that standards of truth in the natural sciences have undergone change; there was a Copernican revolution in astronomy, a Darwinian revolution in biology, an Einsteinian revolution in physics. But do scientific revolutions like these occur in mathematics? Mathematicians have most often answered this question as did the nineteenth-century mathematician Hermann Hankel, who said, "In most sciences, one generation tears down what another has builf, and what one has established, the next undoes. In mathematics alone, each generation builds a new story to the old structure." [20, p Hankel's view is not, however, completely valid. There have been several major upheavals in mathematics. For example, consider the axiomatization of' geometry in ancient Greece, which transformed mathematics from an experimental science into a wholly intellectual one. Again, consider the discovery of non-euclidean geometries

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