# Real-valued function of one variable which is continuous on [a,b] and semi-differentiable on [a,b)?

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There does not exist such a function, even if you omit the assumption of continuity. For an arbitrary function, if the left and right derivatives exist at each point (even when $$+\infty$$ and $$-\infty$$ are allowed as values), then these unilateral derivatives differ on an at most countable set. This was proved for continuous functions by the Italian mathematician Beppo Levi in 1906 [9]. Also in 1906, Hilbert [5] essentially proved the same result in the course of an unrelated investigation. I believe what Hilbert did was to prove the result "in passing" for certain special continuous functions that arose in his investigation, with a proof that essentially applies to an arbitrary continuous function. Unaware of Levi's or Hilbert's results, Rosenthal [10] and Sierpiński [11] independently proved this result for continuous functions in 1912. On p. 148 of Young [13] there is a brief discussion of Hilbert's result and, in footnote 1, Rosenthal's and Sierpiński's results. Sierpiński [11] also gave an example of a continuous function such that each unilateral derivative exists and is finite at each point, and such that these unilateral derivatives differ on a set that is dense in $$\mathbb R$$. (The details of Sierpiński's example were published in Polish in another 1912 paper, JFM 43.0482.02.)

A stronger and more general result was published by Grace Chisholm Young in 1914 [13]. Young was aware of Levi's and Hilbert's results, but not those of Rosenthal and Sierpiński, at least not until Young's paper was essentially complete.

In what follows, $$D^{-}f$$ and $$D^{+}f$$ denote the upper left and upper right Dini derivates of $$f$$ and $$D_{-}f$$ and $$D_{+}f$$ denote the lower left and lower right Dini derivates of $$f.$$

Theorem (G. C. Young, 1914): Let $$f: {\mathbb R} \rightarrow {\mathbb R}$$ be an arbitrary function. Then the following set is countable:

$$\left\{c \in {\mathbb R}: \; D^{+}f(c) < D_{-}f(c) \right\} \cup \left\{c \in {\mathbb R}: \; D^{-}f(c) < D_{+}f(c) \right\}$$

To better understand what this says, let $${\mathcal D}^{-}(f,c)$$ denote the set of extended real number subsequential limits of $$\frac{f(x) - f(c)}{x-c}$$ as $$x \rightarrow c^{-},$$ and let $${\mathcal D}^{+}(f,c)$$ denote the set of extended real number subsequential limits of $$\frac{f(x) - f(c)}{x-c}$$ as $$x \rightarrow c^{+}.$$ Then Young's 1914 theorem says that there exist at most countably many values of $$c$$ such that one of these sets, $${\mathcal D}^{-}(f,c)$$ or $${\mathcal D}^{+}(f,c),$$ lies entirely to the left of the other set on the extended real number line (i.e. each point in one of these sets lies to the left of each point in the other set). Incidentally, both $${\mathcal D}^{-}(f,c)$$ and $${\mathcal D}^{+}(f,c)$$ are closed (in fact, compact) subsets of the extended real line such that $$D_{-}f(c)$$ and $$D^{-}f(c)$$ are the least and greatest elements in $${\mathcal D}^{-}(f,c)$$, and $$D_{+}f(c)$$ and $$D^{+}f(c)$$ are the least and greatest elements in $${\mathcal D}^{+}(f,c).$$ When $$f$$ is continuous, $${\mathcal D}^{-}(f,c)$$ and $${\mathcal D}^{+}(f,c)$$ are the closed intervals $$[D_{-}f(c),\;D^{-}f(c)]$$ and $$[D_{+}f(c),\;D^{+}f(c)]$$ in the extended real line. (Regarding this last sentence, it will be instructive to consider, for each real number $$-1 \leq m \leq 1$$, the intersections of the line given by $$y = mx$$ with the graph of $$f(x) = x \sin \left(\frac{1}{x}\right)$$ with $$f(0)=0$$.)

Proof: Let $$E = \left\{c \in {\mathbb R}: \; D^{+}f(c) < D_{-}f(c)\right\}$$. We will show that $$E$$ is countable. A similar argument can be used to show the countability of the set of points such that $$D^{-}f(c) < D_{+}f(c)$$ (or simply apply the result we prove to the function $$f(-x)$$). For each pair of rational numbers $$r,s$$ such that $$r < s,$$ and for each positive integer $$n,$$ let

$$E_{rsn} \;\; = \;\; \left\{c \in E: \; c < x < c + \frac{1}{n} \; \implies \; \frac{f(x) - f(c)}{x-c} < r \right\} \; \cap \; \left\{c \in E: \; c - \frac{1}{n} < x < c \; \implies \; \frac{f(x) - f(c)}{x-c} > s \right\}$$

Then, as one can verify, we have:

(1) $$E \; = \; \cup \left\{E_{rsn}: \; r \in {\mathbb Q}, \; s \in {\mathbb Q}, \; r < s, \; n \in {\mathbb N} \right\}$$

(2) Each of the sets $$E_{rsn}$$ has the property that its intersection with every open interval of length $$\frac{1}{n}$$ contains at most one point. That is, each of the sets $$E_{rsn}$$ is $$\frac{1}{n}$$-isolated.

From (2) it follows that each of the sets $$E_{rsn}$$ is countable. Hence, from (1) it then follows that $$E$$ is a countable union of countable sets, which completes the proof.

I've adapted this proof from the proof in Bruckner's book [2] (Theorem 4.1, pp. 45-46). Proofs can also be found in Hobson [6] (Article 292, pp. 392-393) and Kannan/Krueger [7] (Theorem 3.5.4, pp. 68-69). Blumberg [1] considered various generalizations from a certain unified point of view, a view that Bruckner/Goffman's paper [3] introduced to many present-day researchers. A number of rather extensive generalizations of this result can be found in Chapter 6 of Thomson's book [12] (especially pp. 143-147).

The theorem above implies that, for an arbitrary function, if both unilateral derivatives exist at each point, then these unilateral derivatives differ on an at most countable set. The reason is that in this case the sets $${\mathcal D}^{-}(f,c)$$ and $${\mathcal D}^{+}(f,c)$$ are the singleton sets $$\left\{f'_{-}(c)\right\}$$ and $$\left\{f'_{+}(c)\right\},$$ and so the property that one of the sets $${\mathcal D}^{-}(f,c)$$ or $${\mathcal D}^{+}(f,c)$$ lies entirely to the left of the other set on the extended real number line reduces to the property that $$f'_{-}(c) \neq f'_{+}(c).$$ However, the theorem above tells us nothing about the set of points where one of the unilateral derivatives exists and the other doesn't exist since, for example, this situation does not require that the unilateral derivative $$f'_{+}(c)$$ lies entirely to the left of the set $${\mathcal D}^{-}(f,c)$$ or entirely to the right of the set $${\mathcal D}^{-}(f,c).$$

Bruckner [2] (Example 4.3, pp. 46-47) gives an example (a "slight variant" of an example given in Leonard [8]) of a continuous function such that there exist continuum many points $$c$$ such that $$f'_{+}(c) = 0$$ and $${\mathcal D}^{-}(f,c) = [-\infty, 0].$$ Although the set of these points is maximal with respect to possible cardinality, the set is small in other respects. For example, the set in question (the bilateral limit points of a fairly thin Cantor set) is nowhere dense (hence, meager) and measure zero. Belna/Cargo/Evans/Humke [4] (Theorem 4, p. 259) show that any such set must be $$\sigma$$-porous (hence, both meager and measure zero), at least for finite unilateral derivatives. More precisely, they show that the set of points at which an arbitrary function has a finite unilateral derivative and isn't differentiable is a $$\sigma$$-porous subset of $$\mathbb R$$. Independently, Zajicek [14] (Corollary on p. 202) proved the same result.

[1] Henry Blumberg, A theorem on arbitrary functions of two variables with applications, Fundamenta Mathematicae 16 (1930), 17-24. JFM 56.0920.01

http://matwbn.icm.edu.pl/ksiazki/fm/fm16/fm1613.pdf

[2] Andrew Michael Bruckner, Differentiation of Real Functions, 2nd edition, CRM Monograph Series #5, American Mathematical Society, 1994, xii + 195 pages. MR 94m:26001; Zbl 796.26001 [The 1st edition was published in 1978 as Springer-Verlag's Lecture Notes in Mathematics #659. The 2nd edition is essentially unchanged from the 1st edition with the exception of a new chapter on recent developments (23 pages) and 94 additional bibliographic items.]

[3] Andrew Michael Bruckner and Casper Goffman, The boundary behavior of real functions in the upper half plane, Revue Roumaine de Mathématiques Pures et Appliquées 11 (1966), 507-518. MR 34 #5995; Zbl 171.30201

[4] Charles Leonard Belna, Gerald Thomas Cargo, Michael Jon Evans, and Paul Daniel Humke, Analogues of the Denjoy-Young-Saks theorem, Transactions of the American Mathematical Society 271 #1 (May 1982), 253-260. MR 83f:26005; Zbl 486.26003

[5] David Hilbert, Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen (Vierte Mitteilung) [Main features of a general theory of linear integral equations (Fourth Communication)], Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse, 1906, 157-228. JFM 37.0351.03

[6] Ernest William 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

[7] Rangachary Kannan and Carole King Krueger, Advanced Analysis on the Real Line, Universitext Series, Springer-Verlag, 1996, x + 259 pages. MR 97m:26001; Zbl 855.26001

[8] John Lander Leonard, Some conditions implying the monotonicity of a real function, Revue Roumaine de Mathématiques Pures et Appliquées 17 (1972), 757-780. MR 46 #3709; Zbl 239.26011

[9] Beppo Levi, Richerche sulle funzioni derivate [Research on derived functions], Atti della Accademia Reale (Nazionale) dei Lincei, Rendiconti, Classe di Scienze fisiche, Matematiche e Naturali (5) 15 (1st semester) (1906), 433-438 & 551-558 & 674-684. JFM 37.0311.04

[10] Artur Rosenthal, Über die singularitäten der reellen ebenen kurven [On the singularities of real plane curves], Mathematische Annalen 73 (1913), 480-521. JFM 44.0642.06 [Published version of Rosenthal's 1912 Habilitation Thesis (JFM 43.0649.02).]

http://tinyurl.com/7aqu5hr

[11] Waclaw Franciszek Sierpiński, Sur l'ensemble des points angulaires d'une courbe $$y = f(x)$$ [On the angular points of the curve $$y = f(x)$$], Bulletin de l'Académie des Sciences Cracovie, Classe des Sciences Mathématiques et Naturelles, Série A (1912), 850-855. JFM 43.0650.01 [Reprinted on pp. 67-71 of Volume 3 of Sierpiński's Oeuvres Choisies.]

[12] Brian S. Thomson, Real Functions, Springer-Verlag, Lecture Notes in Mathematics #1170, viii + 229 pages. MR 87f:26001; Zbl 581.26001

[13] Grace Chisholm Young, A note on derivates and differential coefficients, Acta Mathematica 37 (1914), 141-154. JFM 45.0433.10

http://www.kryakin.com/files/Acta_Mat_(2_55)/acta56_1/37/37_05.pdf

[14] Ludek Zajicek, On the symmetry of Dini derivates of arbitrary functions, Commentationes Mathematicae Universitatis Carolinae 22 (1981), 195-209. MR 82c:26009; Zbl 462.26003

http://dml.cz/bitstream/handle/10338.dmlcz/106064/CommentatMathUnivCarol_022-1981-1_16.pdf

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Updated on August 01, 2022