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  • Roger Bagula
    http://groups.google.com/group/sci.math/msg/814be41b1ea8c024 Dave L. Renfro Jan 23 2000, 1:00 am show options
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      http://groups.google.com/group/sci.math/msg/814be41b1ea8c024
      Dave L. Renfro  Jan 23 2000, 1:00 am     show options
      Newsgroups: sci.math
      From: dlren...@... (Dave L. Renfro) - Find messages by this author
      Date: 2000/01/23
      Subject: HISTORICAL ESSAY ON CONTINUITY OF DERIVATIVES
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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.

      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
      -- 
      Roger L. Bagula { email: rlbagula@...  or  rlbagulatftn@... }                             
      11759 Waterhill Road,                               
      Lakeside, Ca. 92040    telephone: 619-561-0814
      
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