Bibliography for Singular Functions [closed]

Question

I wound up assembling a rather lengthy and partially annotated bibliography for my answer to the math StackExchange question Singular continuous functions, but it seems I got a little too carried away and found myself with a post well over the maximum of 30,000 characters. Thus, I'm using this "question" to post the full bibliography, which I think could be a useful reference to others.

I am including a large portion of the bibliography in the question because I still have too many characters otherwise.

SOME HISTORY

It is easy to construct non-decreasing continuous functions with a zero derivative almost everywhere, and such functions have numerous appearances in the literature beginning around 1883/1884 in papers by Cantor and Scheeffer (separately authored). See Fleron (1994) [27] and Maurey/Tacchi (2002) [54].

On the other hand, the first proof of the existence of strictly increasing continuous functions with a zero derivative almost everywhere seems to be in Ernst Hellinger's 1907 Ph.D. Dissertation, but I haven't looked into this to see exactly what he proved and where in the Dissertation it can be found. (Minkowski's function came earlier, but it wasn't proved to have this property until Denjoy proved it in 1934/1938 [17] [18].) Likely unaware of Hellinger's result, Denjoy gave such a function in 1915 [15]. Other early examples of such functions are given in Sierpinski (1916; cites Denjoy) [73], Hahn (1921; cites no one) [32], Rajchman (1921; cites Denjoy and Sierpinski) [64], Vitali (1922; I have not seen this paper) [77], and Blumberg (1926; cites no one) [2]. Blumberg's example is the first such function I know of that appeared in English. Curiously, I found this seemingly straightforward historical issue of priority especially difficult to uncover. Other than items published recently (last 10 years or so), the only explicit statement of priority I've seen is Salem (1943, p. 427, footnote) [68], who credits Denjoy with giving the first example.

BIBLIOGRAPHY FOR SINGULAR FUNCTIONS

[1] Giedrius Alkauskas, Extensive bibliography on the Minkowski Question Mark Function and allied topics, internet web page, 63 entries. Accessed 15 February 2014.

[2] Henry Blumberg, Non-measurable functions connected with certain functional equations, Annals of Mathematics (2) 27 #3 (March 1926), 199-208.

In a Note (pp. 206-208) at the end of the paper Blumberg constructs a function that he then proves is strictly increasing, continuous, and has a zero derivative almost everywhere. This particular appearance (in English!) of such a function seems to have been missed by authors who cite early appearances of such functions. A footnote on the first page says that the paper was read "in part" on 1 December 1917, so it is possible that Blumberg was in possession of this example back in 1917. However, the footnote does explicitly say in part, and the abstract for this paper in Bull. Amer. Math. Soc. 24 #5 (February 1918) [see abstract #8 on p. 220] does not mention the example, so in my opinion the evidence is not very strong that Blumberg was in possession of the example in 1917.

[3] Ralph Philip Boas, Oscillating functions, Duke Mathematical Journal 5 #2 (June 1939), 394-400.

Various Baire category results (in various function spaces) relating to the properties of bounded variation, absolute continuity, and having a bounded derivative.

[4] Ralph Philip Boas, Differentiability of jump functions, Colloquium Mathematicum 8 #1 (1961), 81-82.

This paper gives an elementary proof (no use of differentiating monotonic functions; no use of integration; no use of Lebesgue density) that the derivative of a non-decreasing jump function is zero almost everywhere. See Lipinski (1957, 1961), Marczewski (1955), Pu/Pu (1983), Pu (1980-81), Piranian (1966), and Rubel (1963).

The relevance of these papers here is that any non-decreasing function $f$ can be written as $f = A + S + J,$ where $A$ is non-decreasing and absolutely continuous, $S$ is non-decreasing and continuous and has a zero derivative almost everywhere, and $J$ is a non-decreasing jump function (also called a saltus function). This decomposition is unique up to the addition or subtraction of constant functions to $A,$ $S,$ and $J.$ Roughly speaking, this is a decomposition of the function $f$ into a continuous function $A$ that is "maximally nice" for certain integration purposes, a continuous function $S$ that is "maximally bad" for certain integration purposes, and a function $J$ that is the "discontinuous part" of the original function $f.$

An absolutely continuous function is continuous and also nice in many other ways. For example, an absolutely continuous function is finitely differentiable almost everywhere. Of course, any non-decreasing function is also finitely differentiable almost everywhere, but this property is true for absolutely continuous functions even when no additional monotonicity condition is assumed. By way of contrast, recall that a continuous function can be nowhere differentiable. An absolutely continuous function $A$ also satisfies the following version of the Fundamental Theorem of Calculus: $\int_{a}^{b}A'(x)dx = A(b) - A(a)$ (Lebesgue integration).

A singular function $S$ fails to satisfy this Fundamental Theorem of Calculus version in the worst way: For any interval $[a,b]$ on which $S$ is defined, we have $\int_{a}^{b}S'(x)dx = 0$ (Lebesgue integration).

A jump function is the sum of a constant function and a function that can be defined in the following way. Let $Z$ be a nonempty countable (finite or infinite) set of real numbers and let $B = \{b_{z}: \; z \in Z\}$ be a set of positive real numbers such that $Z$ and $B$ have the same number of elements (same cardinality) and the sum of all the numbers in $B$ is finite (of course, this is automatic if $B$ is a finite set). For each $z \in Z$ define the function $f_z$ by $f_{z}(x) = 0$ if $x < z$ and $f_{z}(x) = b_z$ if $x \geq z.$ Finally, a jump function is (any constant function added to) the zero function or a function that equals $\sum_{z \in Z}f_{z}$ for some choice of the sets $Z$ and $B.$ It can be shown that such a function $J$ is continuous at each point not belonging to $Z,$ and $J$ is discontinuous at each $z \in Z$ with $\limsup_{x \rightarrow z}J(x) - \liminf_{x \rightarrow z}J(x) = b_{z}.$

[5] Andrew Michael Bruckner and John Lander Leonard, On differentiable functions having an everywhere dense set of intervals of constancy, Canadian Mathematical Bulletin 8 #1 (February 1963), 73-76.

Let $P$ be a perfect nowhere dense subset of $[0,1]$ (i.e. $P$ is a Cantor set). They prove the following result. There exists a function $f:[0,1] \rightarrow \mathbb R$ that (i) $f$ is constant on each open interval contiguous to $P$ $(f$ can have different constant values on different contiguous intervals) AND (ii) $f$ is not constant on each open interval containing a point of $P$ AND (iii) $f$ is finitely differentiable on $(0,1)$ IF AND ONLY IF $P$ has the property that the intersection of $P$ with every open subinterval of $[0,1]$ has positive Lebesgue measure (i.e. $P$ is "measure dense", also called "metrically dense"). This has the following implication for the Cantor staircase function $F$ (defined by making use of the usual Cantor middle thirds set that has measure zero). No matter how we redefine the values of $F$ at points in the Cantor set, it is not possible to change (or to "smooth out") the continuous function $F$ in such a way that we will get a finitely differentiable function, even though $F$ has a zero derivative almost everywhere.

[6] Frank Sydney Cater, Most monotone functions are not singular, American Mathematical Monthly 89 #7 (Aug.-Sept. 1982), 466-469.

Complements Zamfirescu (1981). See MR 92g:26015 and the references there for more elaborate and generalized versions.

[7] Frank Sydney Cater, Mappings into sets of measure zero, Rocky Mountain Journal of Mathematics 16 #1 (Winter 1986), 163-171.

[8] Lamberto Cesari, Variation, multiplicity, and semicontinuity, American Mathematical Monthly 65 #5 (May 1958), 317-332.

A lengthy and detailed expository survey of classical results involving variation and absolute continuity of functions.

[9] Donald Richard Chalice, A characterization of the Cantor function, American Mathematical Monthly 98 #3 (March 1991), 255-258.

The following theorem is proved (italics not in original): "Any real-valued function $F(x)$ on $[0,1]$ that is monotone increasing and satisfies (a) $F(0)=0,$ (b) $F(x/3)=F(x)/2,$ and (c) $F(1-x) = 1 - F(x),$ is the Cantor function.

[10] Sandra Lynn Cousins, Singular functions, Pi Mu Epsilon Journal 7 #6 (Spring 1982), 374-381.

An elementary expository survey written by an undergraduate for the national (U.S.) Pi Mu Epsilon Student Paper competition (won "Second Prize"). Includes detailed discussions of the Cantor staircase function and Hellinger's function.

[11] Richard Brian Darst, Some Cantor sets and Cantor functions, Mathematics Magazine 45 #1 (January 1972), 2-7.

This is an expository paper. Darst begins with a discussion of arc length (defined as the least upper bound of the lengths of polygonal paths with vertices on the graph), then the Cantor middle thirds set is described, then the Cantor staircase function from $[0,1]$ onto $[0,1]$ is described and shown to have length $2$ and shown to have no finite two-sided derivative at any point of the Cantor middle thirds set. In the last section of the paper (pp. 5-7), Darst considers Cantor sets defined by removing (from the various closed intervals at each construction stage) centrally located open intervals whose lengths are a fixed proportion $0 < \lambda \leq 1$ of those removed when constructing the Cantor middle thirds set. When $\lambda = 1$ we obtain the Cantor middle thirds set, and when $0 < \lambda < 1$ we obtain a Cantor set with Lebesgue measure $1 - \lambda.$ Darst then shows that the length of the corresponding Cantor function from $[0,1]$ onto $[0,1]$ is $\lambda + \sqrt{1 + (1-\lambda)^{2}}$ and, when $\lambda < 1,$ Darst shows that the two-sided derivative of the corresponding Cantor function is $\frac{1}{1 - \lambda}$ at almost every point of the Cantor set associated with $\lambda.$ (Here, "almost every point" means the complement of the set of points has Lebesgue measure zero.)

[12] Richard Brian Darst, The Hausdorff dimension of the nondifferentiability set of the Cantor function is $[\ln(2)/\ln(3)]^2$, Proceedings of the American Mathematical Society 119 #1 (September 1993), 105-108.

See Falconer (2004).

[13] Richard Brian Darst, Hausdorff dimension of sets of non-differentiability points of Cantor functions, Mathematical Proceedings of the Cambridge Philosophical Society 117 #1 (January 1995), 185-191.

[14] Frederik Michel Dekking and Wenxia Li, How smooth is a Devil's staircase?, Fractals 11 #1 (March 2003), 101-107.

[15] Arnaud Denjoy, Mémoire sur les nombres dérivés des fonctions continues [Memoir on the derived numbers of continuous functions], Journal de Mathématiques Pures et Appliquées (7) 1 (1915), 105-240.

In Article 63 (Exemple VI, pp. 204-209) Denjoy defines a function that he then proves is continuous (see footnote on p. 208), strictly increasing, and has a zero derivative almost everywhere. However, Denjoy's proof makes use of some of the specialized results that he had previously developed in the paper. Sierpinski (1916) was written in order to give an example of such a function, and a verification of these properties, that only makes use of elementary results in real analysis.

[16] Arnaud Denjoy, Sur quelques points de la théorie des fonctions [On some points in the theory of functions], Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences (Paris) 194 (1932), 44-46.

[17] Arnaud Denjoy, Sur une fonction de Minkowski [On the function of Minkowski], Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences (Paris) 198 (1934), 44-47.

[18] Arnaud Denjoy, Sur une fonction réelle de Minkowski [On the real function of Minkowski], Journal de Mathématiques Pures et Appliquées (9) 17 (1938), 105-151.

[19] Arnaud Denjoy, Propriétés différentielles de la fonction Minkowskienne réelle. Statistique des fractions continues [Differential properties of the real Minkowski function. Statistics of continued fractions], Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences (Paris) 242 (1956), 2075-2079.

[20] Oleksiy [Aleksei] Alfredovich Dovgoshey, Olli Tapani Martio, Vladimir Illich Ryazanov, and Matti Vuorinen, The Cantor function, Expositiones Mathematicae 24 #1 (2006), 1-37.

A very highly recommended survey paper.

[21] Thomas Paul Dence, Differentiable points of the generalized Cantor function, Rocky Mountain Journal of Mathematics 9 #2 (Spring 1979), 214-217.

[22] Harry Dym, On a class of monotone functions generated by ergodic sequences, American Mathematical Monthly 75 #6 (June-July 1968), 594-601.

[23] John [Jack] Alan Eidswick, A characterization of the nondifferentiability set of the Cantor function, **Proceedings of the American Mathematical Society 42 #1 (January 1974), 214-217.

[24] Griffith Conrad Evans, Calculation of moments for a Cantor-Vitali function, American Mathematical Monthly 64 #8 (Part II) (October 1957), 22-27.

See Dovgoshey/Martio/Ryazanov/Vuorinen (2006, Section 5) and Wall (1961).

[25] Kenneth John Falconer, One-sided multifractal analysis and points of non-differentiability of devil's staircases, Mathematical Proceedings of the Cambridge Philosophical Society 136 #1 (January 2004), 167-174.

Author's Abstract (italics not in original): We examine the multifractal spectra of one-sided local dimensions of Ahlfors regular measures on R. This brings into a natural context a curious property that has been observed in a number of instances, namely that the Hausdorff dimension of the set of points of non-differentiability of a self-affine 'devil's staircase' function is the square of the dimension of the set of points of increase. See Darst (1993).

[26] Julian Frederic Fleron, Pointwise Derivates of the Cantor Function, Master of Arts Thesis (under Max August Jodeit), University of Minnesota, June 1990, iv + 106 pages.

This Thesis gives a very thorough expository analysis of Eidswick (1974) and the appropriate background material. TABLE OF CONTENTS: Chapter 1. Introduction §1. Introduction (pp. 1-3); §2. History of the Cantor Set and Cantor Function (pp. 3-8); §3. Representations of the Cantor Set and Cantor Function (pp. 9-17). Chapter 2. Dini Derivates of the Cantor Function §1. Upper and Lower Right Derivates [of the Cantor Function] (pp. 18-33); §2. Bounding the Upper and Lower Left Derivates [of the Cantor Function] (pp. 33-39); §3. Upper and Lower Left Derivates [of the Cantor Function] (pp. 39-43); §4. Finite Derivates [of the Cantor Function] (pp. 43-51). Chapter 3. Sets of Derivates §1. Preliminary Investigation of the Ternary Structure (pp. 52-56); §2. Hausdorff Measure [of Some Previously Studied Sets] (pp. 56-63); §3. Hausdorff Dimension and Category [and Density of Some Previously Studied Sets] (pp. 63-66). Chapter 4. The Ternary Structure and Sets of Derivates §1. Further Investigation of the Ternary Structure (pp. 67-80). §2. Approximating a Special Essential Secant (pp. 80-85); §3. Constructing Points with Arbitrary Lower Derivates (pp. 85-95). Appendix §1. Construction of a Point Where $f_{+}f(x) = a$ (pp. 96-98); §2. Hausdorff Dimension of the Cantor Set (pp. 98-102).

[27] Julian Frederic Fleron, A note on the history of the Cantor set and Cantor function, Mathematics Magazine 67 #2 (April 1994), 136-140.

[28] Gerald Freilich, Increasing continuous singular functions, American Mathematical Monthly 80 #8 (October 1973), 918-919.

Let $C(x)$ be the Cantor staircase function and let $\{a_{1},\,a_{2},\, \ldots\}$ be a countable set that is dense in ${\mathbb R}.$ Freilich gives a short proof that $\sum_{n=1}^{\infty}2^{-n}C\left(2^{n}(x-a_{n})\right)$ is continuous, strictly increasing, and has a zero derivative almost everywhere. Fubini's theorem on differentiation of series is used to establish "zero derivative almost everywhere".

[29] Krishna Murari Garg, On singular functions, Revue Roumaine de Mathématiques Pures et Appliquées 14 #10 (1969), 1441-1452. Zbl 191.34601 review

[30] Krishna Murari Garg, Construction of absolutely continuous and singular functions that are nowhere of monotonic type, pp. 61-79 in Daniel Waterman (editor), Classical Real Analysis, Contemporary Mathematics 42 (1985), x + 216 pages.

[30] Krishna Murari Garg, Construction of absolutely continuous and singular functions that are nowhere of monotonic type, pp. 61-79 in Daniel Waterman (editor), Classical Real Analysis, Contemporary Mathematics 42 (1985), x + 216 pages.

[31] Ray Edwin Gilman, A class of functions continuous but not absolutely continuous, Annals of Mathematics (2) 33 #3 (July 1932), 433-442.

This is a sequel to Hille/Tamarkin (1929) in which the base 3 representation into a base 2 representation is generalized to base $\alpha$ representation into a base $\beta$ representation, where $\alpha$ and $\beta$ are integers such that $1 < \beta < \alpha$ and $\beta - 1$ divides $\alpha - 1.$ Also, there is more focus in Gilman's paper on the Dini derivate behavior of the corresponding functions than there is in Hille/Tamarkin's paper. See Dovgoshey/Martio/Ryazanov/Vuorinen (2006, p. 32, Remark 10.4) for an erroneous claim in Gilman's paper.

[32] Hans Hahn, Theorie der Reellen Funktionen [Theory of Real Functions], Verlag von Julius Springer (Berlin), 1921, viii + 600 pages.

Hahn gives an example of a strictly increasing continuous function with a zero derivative almost everywhere on pp. 538-539.

[33] Philip Hartman and Richard Brandon Kershner, The structure of monotone functions, American Journal of Mathematics 59 #4 (October 1937), 809-822.

This is a study of the absolutely continuous and singular behavior of continuous non-decreasing functions $f: [0,1] \rightarrow [0,1]$ "in terms of the asymptotic or qualitative properties of the two dense sequences of numbers which are mapped on to each other by $y = f(x).$" Section 5 (pp. 818-819) proves a general result involving modulus of continuity that implies (as a special case) for each $0 < \alpha < 1$ there exists a strictly increasing continuous function $f$ with derivative zero almost everywhere such that $f$ has Lipschitz order exactly $\alpha$.

Answer 1

[34] Fritz Herzog and Barnard Hinkle Bissinger, A Cantor function constructed by continued fractions, Bulletin of the American Mathematical Society 53 #2 (February 1947), 104-115.

Similar to Gilman (1932), except the representations of the Cantor sets makes use of continued fraction considerations instead of $\alpha$-ary and $\beta$-ary expansion considerations.

[35] Carl Einar Hille and Jacob David Tamarkin, Remarks on a known example of a monotone continuous function, American Mathematical Monthly 36 #5 (May 1929), 255-264.

This is an expository survey of the Cantor middle thirds staircase function. See Gilman (1932).

[36] Egbert Rudolf van Kampen and Aurel Friedrich Wintner, On a singular monotone function, Journal of the London Mathematical Society (1) 12 #4 (October 1937), 243-244.

Construction of a strictly increasing continuous function with a zero derivative almost everywhere and a proof that, for the function constructed, there exists a set of full measure whose image under the function has measure zero. The authors observe that the result is known, but state that their construction "seems to be simpler and more direct than the usual procedures".

[37] Rangachary Kannan and Carole King Krueger, Advanced Analysis on the Real Line, Universitext series, Springer-Verlag, 1996, x + 259 pages.

This book contains a lot of material on bounded variation, absolute continuity, and singular functions: Chapter 6: Bounded Variation (pp. 118-152; 29 chapter exercises); Chapter 7: Absolute Continuity (pp. 153-180; 22 chapter exercises); Chapter 8: Cantor Sets and Singular Functions (pp. 181-215; 23 chapter exercises); Chapter 9: Spaces of BV and AC Functions (pp. 216-245; no chapter exercises).

[38] Kiko Kawamura, The derivative of Lebesgue's singular function, Real Analysis Exchange, Summer Symposium 2010, 83-85.

[39] Kiko Kawamura, On the set of points where Lebesgue's singular function has the derivative zero, Proceedings of the Japan Academy 87(A) #9 (2011), 162-166.

From the first page (italics not in original): "It is well-known that $L_{a}(x)$ is strictly increasing, but the derivative is zero almost everywhere. See Fig. 1 for the gragh [sic] of $L_{a}(x).$ This distribution function $L_{a}(x)$ was also defined in different ways and studied by a number of authors: Cesaro (1906), Faber (1910), Lomnicki and Ulam (1934), Salem (1943), De Rham (1957) and others. $[\cdots ]$ Reconsider the differentiability of $L_{a}(x).$ It is known that for any $x \in [0,1], \; L'_{a}(x)$ is either zero, or $+\infty,$ or it does not exist. Then, it is natural to ask at which points $x \in [0,1]$ exactly [do] we have $L'_{a}(x) = 0$ or $+\infty.$

[40] Alexander B. Kharazishvili, Strange Functions in Real Analysis, Monographs and Textbooks in Pure and Applied Mathematics #229, Marcel Dekker, 2000, viii + 297 pages.

See Chapter 2: Singular monotone functions (pp. 55-68). Roughly, the topics covered in this chapter are the following. A proof that each monotone function has at most countably many points of discontinuity, each countable set is the discontinuity set for some strictly increasing continuous function, each monotone function is finitely differentiable almost everywhere, Exercise 5 (p. 62) outlines the construction of a continuous non-decreasing function with a two-sided infinite derivative at each point of any specified measure zero set [Incidentally, given any real-valued Lebesgue measurable function $f,$ the set $\{x: \; f'(x) = +\infty\}$ (and likewise the set where $f'(x) = -\infty$) has Lebesgue measure zero. This was first proved for continuous functions by Luzin in 1912, and has since been extensively extended into what today is known as the Denjoy-Young-Saks theorem.], Fubini's differentiation of series theorem, construction of a strictly increasing continuous function with a zero derivative almost everywhere (pp. 66-68). As far as I can tell, there is no study of the Dini derivates of such functions (either Cantor staircase functions or strictly increasing singular functions) in this chapter or elsewhere in the book.

[41] John Rankin Kinney, Singular functions associated with Markov chains, Proceedings of the American Mathematical Society 9 #4 (August 1958), 603-608.

[42] John Rankin Kinney, Note on a singular function of Minkowski, Proceedings of the American Mathematical Society 11 #5 (October 1960), 788-794.

[43] Walter Eugene Klann, Properties and Applications of Absolutely Continuous Functions, Ed.D. Dissertation (under Donald Dale Elliott), Colorado State College [= University of Northern Colorado], 1968, ix + 190 + 1 pages.

This is an extremely well written, well informed, and thorough survey (93 item bibliography) of its topic up to the early 1960s. Although singular functions only play a minor role, I am including this reference because of its outstanding expository quality and because it is likely to be virtually unknown (since it is not a mathematics Ph.D. Dissertation). TABLE OF CONTENTS: Chapter I: Introduction (pp. 1-4). Chapter II: Review of Related Literature (pp. 5-8). Chapter III: Fundamental Properties (pp. 9-33). Topics in this chapter: The Cantor Function, Total Variation Properties, Differentiation Properties. Chapter IV. The Role of Absolutely Continuous Functions in Lebesgue Integration (pp. 34-74). Topics in this chapter: The Banach-Zarecki Theorem, Indefinite Integral of a Function, Relation of Stieltjes and Lebesgue Integrals, Integration by Parts, Integration by Substitution. Chapter V: Compositions of Absolutely Continuous Functions (pp. 75-158). Topics in this chapter: Conditions $T_1$ and $T_{2},$ Necessary and Sufficient Conditions for Compositions, The Nearly Everywhere condition, Ridée Functions, The Fundamental Theorem, Transfinite Compositions. Chapter VI: Arc Length, Sequences, and Recent Applications (pp. 159-183). Topics in this chapter: Total Variation of the Function $f(x+u)-f(x)$, Convergence of Sequences, Opial's Inequality.

[44] Hermann Kober, On singular functions of bounded variation, Journal of the London Mathematical Society (1) 23 #3 (July 1948), 222-229.

Among other things, Kober proves the following. Let $f$ a function of bounded variation on $[a,b].$ Then $f$ has a zero derivative almost everywhere in $[a,b]$ if and only if the length of the curve $y = f(x)$ from $x=a$ to $x=b$ is $b - a + V(f,a,b),$ where $V(f,a,b)$ is the variation of $f$ on the interval $[a,b].$ It follows that for a function that is non-decreasing on $[a,b],$ having a zero derivative almost everywhere is equivalent to having length $b - a + f(b) - f(a)$ between $x=a$ and $x=b.$ (See also MR 50 #4859.) Thus, the well known fact that the graphs of singular continuous functions from $[0,1]$ onto $[0,1]$ have length $2$ (the maximum possible length for a non-decreasing function from $[0,1]$ to $[0,1]$) is highly dependent on the singular nature of the function. See Mauldon (1966) and Zaanen/Luxemburg (1963).

[45] Hermann Kober, A remark on a monotone singular function, Proceedings of the American Mathematical Society 3 #3 (June 1952), 425-427.

Generalizes some of the results in Kober (1948).

[46] Wenxia Li, Non-differentiability points of Cantor functions, Mathematische Nachrichten 280 #1-2 (January 2007), 140-151.

[47] Jan Stanislaw Lipinski, Sur la dérivée d'une fonction de sauts [On the derivative of a saltus function], Colloquium Mathematicum 4 #2 (1957), 197-205.

See Boas (1961) and the references given there.

[48] Jan Stanislaw Lipinski, Une simple démonstration du théorème sur la dérivée d'une fonction de sauts [A simple proof of the theorem on the derivative of a saltus function], Colloquium Mathematicum 8 #2 (1961), 251-255.

See Boas (1961) and the references given there.

[49] Jan Stanislaw Lipinski, On derivatives of singular functions, Bulletin de l'Académie Polonaise des Sciences. Série des Sciences Mathématiques, Astronomiques et Physiques 20 #8 (1972), 625-628.

Author's Abstract (italics not in original): This work deals with singular functions defined on the interval $<0,1>.$ Among others it is proved that for every singular function $f$ the set $\{x: \; f'(x) = +\infty\}$ is a subset of a set of type $F_{\sigma}$ and of measure zero. It is also proved that for every set $F,$ if $F \subset E \in F_{\sigma}$ and $E$ is of measure zero, there exists a singular function $f$ such that $F \subset \{x: \; f'(x) = +\infty\}.$ These theorems characterize the sets of points in which a singular function can attain an infinite derivative. Note: Every "subset of a set of type $F_{\sigma}$ and of measure zero" (equivalently, a subset of a countable union of closed measure zero sets) is a measure zero set that is also first Baire category (i.e. a meager set). However, there exist sets that are simultaneously measure zero and first Baire category that cannot be covered by countably many such sets. (See my May 2000 essay for more information and a lot of references.) Thus, the sets where such functions have an infinite derivative are strictly smaller than being simultaneously measure zero and first Baire category. Nonetheless, these sets can have a cardinality continuum intersection with every interval, and even a Hausdorff dimension $1$ intersection with every interval. See Oshime (1994).

[50] Edward Marczewski [Szpilrajn], Uwagi o zbiorach miary zero i o rozniczkowalnosci funkcji monotonicznych [Remarks on sets of measure zero and the derivability of monotonic functions], Roczniki Polskiego Towarzystwa Matematycznego. [Annales Societatis Mathematicae Polonae.] Seria I. Prace Matematyczne 1 (1955), 141-144.

The following is the entire English summary on p. 144 (italics in the original): "Remarks on the set of points of non-derivability of a monotonic function; in particular a proof of the following THEOREM. For every linear set $N$ of measure zero there exists a purely discontinuous monotonic function which is non derivable at every point belonging to $N.$" Note: "linear set" means a subset of $\mathbb R$ and "purely discontinuous monotonic function" means a jump function in the sense that I define above in Boas (1961) (or, in the case of a non-increasing function, the additive inverse of what I defined). See Boas (1961) and the references given there.

[51] José M. Martinez-Blanco, Representaciones de Cantor y Funciones Singulares Asociadas [Representations of Associated Cantor Singular Functions], Ph.D. Dissertation (under Eusebio Corbacho Rosas), Universiad de Vigo (Spain), 1999.

[52] James Grenfell Mauldon, Continuous functions with zero derivative almost everywhere, Quarterly Journal of Mathematics (Oxford) (2) 17 (1966), 257-262.

The main result is a theorem about the inverse (relative to composition of functions) of strictly increasing continuous singular functions that appears to be essentially the same as one of the main results in Kober (1948). However, Kober (1948) is not cited. I do not know if Mauldon was unaware of Kober's paper or if there is something I am overlooking. Theorem 1 (p. 257; italics in original): If $f$ is a real-valued strictly increasing continuous function of a real variable which possesses almost everywhere (a.e.) a derivative with value zero, then the inverse function $f^{-1}$ has the same properties. This theorem is then illustrated in two examples. The first example involves base 3 expansions (i.e. ternary expansions) of real numbers in the interval $\left[-\frac{1}{2}, \, \frac{1}{2}\right]$ and binary expansions of real numbers in the interval $\left[-\frac{1}{3}, \, \frac{1}{3}\right],$ and the notions simply normal to base $3,$ Bernoulli sequences, and Borel's Strong Law of Large Numbers are brought up. The second example involves continued fraction representations and at one point in the second example Mauldon makes the following comment: (p. 261; italics not in original) A direct proof of the fact that the function $f$ has zero derivative almost everywhere is intricate and difficult but, using known results [Khintchine's book Continued Fractions is cited], it is easy to prove as in § 2 that $f^{-1}$ has zero derivative almost everywhere and this, in the light of Theorem 1, is sufficient. In the last section of the paper Mauldon considers continuous functions $f:[0,1] \rightarrow \mathbb R$ with zero derivative almost everywhere and such that for each $y$ in the range of $f$ the sets $f^{-1}(y) = \{x: \; f(x) = y\}$ all have the same cardinality. Any such function that is strictly increasing (such as in Mauldon's two earlier examples) shows that each of the sets $f^{-1}(y)$ can have cardinality $1.$ Mauldon then gives a short argument, making use of a result in Varberg (1965), that no such example exists with "cardinality $1$" replaced by "cardinality $n$" for any positive integer $n > 1.$ Mauldon ends with an outline ("a rather tedious verification" is omitted) that each of the sets $f^{-1}(y)$ can be countably infinite. See Zaanen/Luxemburg (1963).

[53] Jeremy [Jerry] Gregson Morris, The Hausdorff dimension of the nondifferentiability set of a nonsymmetric Cantor function, Rocky Mountain Journal of Mathematics 32 #1 (Spring 2002), 357-370.

[54] Bernard Maurey and Jean-Pierre Tacchi, Qui a inventé la fonction singulière de Cantor? [Who invented the Cantor singular function?], Rendiconti Accademia Nazionale delle Scienze detta dei XL Memorie di Matematica (5) 26 (2002), 29-34.

[55] Francis Joseph Murray, Nullifying functions, Bulletin of the American Mathematical Society 46 #6 (June 1940), 459-465.

[56] Yorimasa Oshime, Examples of functions which are almost everywhere differentiable although their derivatives are nowhere Lebesgue-integrable, Mathematica Japonica 39 #3 (May 1994), 581-594.

Theorem 3 gives an example of a strictly increasing continuous function whose derivative is zero on the complement of a set having Hausdorff dimension zero (a set that is much smaller than a set with measure zero). That is, a function such that $f'(x) = 0$ everywhere except for a Hausdorff dimension zero set. See Lipinski (1972).

[57] Jaume Paradís, Pelegrí Viader, and Lluís Bibiloni, The derivative of Minkowski's $?(x)$ function, Journal of Mathematical Analysis and Applications 253 #1 (1 January 2001), 107-125.

The main result proved is that for each $0 < x < 1$ the Minkowski $?(x)$ function satisfies one of the following: (1) The two-sided derivative equals $0.$ (2) The two-sided derivative equals $\infty .$ (3) The two-sided derivative does not exist, finitely or infinitely. As far as I can tell, the paper does not consider one-sided derivative behavior or Dini derivate behavior. 2nd to last paragraph on p. 114: "These results could lead us to conjecture that this behavior is common to all singular functions but there exist families of singular functions for which there are points in which the derivative is finite and different from zero; see [12]. Note: Their reference [12] is Martinez-Blanco (1999).

[58] George Piranian, The derivative of a monotonic discontinuous function, Proceedings of the American Mathematical Society 16 #2 (April 1965), 243-244.

This gives a relatively short proof that if $E \subseteq {\mathbb R}$ is $G_{\delta}$ and countable (for subsets of $\mathbb R$ this is equivalent to being scattered in the Cantor-Bendixson sense), then there exists a non-decreasing function whose two-sided derivative equals $+\infty$ for each $x \in E$ and equals $0$ for each $x \in {\mathbb R} - E.$ Related is Marczewski (1955), Tolstoff (1940; see MR 2,132a), Bojarski (Annales de la Société Polonaise de Mathématique 24 (1951), pp. 190-191), and Marcus (1962; see MR 26 #2558).

[59] George Piranian, Points of continuity of differentiable jump functions, Revue Roumaine de Mathématiques Pures et Appliquées 11 #8 (1966), 917-919.

See Boas (1961) and the references given there.

[60] Milton Brockett Porter, Concerning absolutely continuous functions, Bulletin of the American Mathematical Society 22 #3 (December 1915), 109-111.

Of possible historical interest. An "inner limiting set" is a $G_{\delta}$ set.

[61] Huay-Min Huoh [Huo Hui Min] Pu and Hwang-Wen Pu, The derivates of the Lebesgue singular function, Tamkang Journal of Mathematics 4 #2 (1973), 45-49. Zbl 287.26008 review

Let $f$ be the Cantor staircase function, let $C$ the Cantor middle thirds set, and let $E$ be the countable collection of endpoints of the open intervals contiguous to $C.$ Pu/Pu prove the following: (i) If $x$ is a right [respectively, left] endpoint of an interval contiguous to $C,$ then $D^{+}f(x) = D_{+}f(x) = \infty$ [respectively, $D^{-}f(x) = D_{-}f(x) = \infty$]. (ii) If $x \in C - E,$ then $D^{+}f(x) = D^{-}f(x) = \infty.$ (iii) Given two points $a < b,$ both in $C,$ there exists points $x_{1} \in C$ and $x_{2} \in C$ such that $a \leq x_{1} < b$ and $a \leq x_{2} < b$ and $D_{+}f(x_{1}) = 0$ and $D_{-}f(x_{2}) = 0.$ (iv) For each $\alpha \geq 0,$ the set $\{x: \; D_{+}f(x) = \alpha\}$ has cardinality continuum and the set $\{x: \; D_{-}f(x) = \alpha\}$ has cardinality continuum.

[62] Huay-Min Huoh [Huo Hui Min] Pu and Hwang-Wen Pu, The derivative of a nondecreasing saltus function, Bulletin of the Institute of Mathematics. Academia Sinica 11 #4 (1983), 505-512.

See the summary in Pu (1980-81). See also See Boas (1961) and the references given there.

[63] Hwang-Wen Pu, On the derivative of a nondecreasing saltus function, Real Analysis Exchange 6 #1 (1980-81), 111-113.

This is a summary of the results in Pu/Pu (1983). See also See Boas (1961) and the references given there.

[64] Aleksander [Alexandre] Rajchman, Une remarque sur les fonctions monotones [A remark on monotone functions], Fundamenta Mathematicae 2 (1921), 50-63.

[65] Gerhard Ramharter, On Minkowski's singular function, Proceedings of the American Mathematical Society 99 #3 (March 1987), 596-597.

[66] Lee Albert Rubel, Differentiability of monotonic functions, Colloquium Mathematicum 10 #2 (1963), 277-279.

See Boas (1961) and the references given there.

[67] Raphaël Salem, On singular monotonic functions of the Cantor type, Journal of Mathematics and Physics (MIT) [later renamed: Studies in Applied Mathematics] 21 (1942), 69-82. Reprinted on pp. 239-251 of Salem's Œuvres Mathématiques (1967).

First sentence of the paper: "This paper is devoted to the study of the Fourier-Stieltjes coefficients of continuous singular monotonic function which are of the Cantor type, that is to say, which are constant in each interval contiguous to a perfect set of measure zero."

[68] Raphaël Salem, On some singular monotonic functions which are strictly increasing, Transactions of the American Mathematical Society 53 #3 (May 1943), 427-439.

From the 2nd paragraph: "While the existence of functions of the Cantor type is almost intuitive and their construction is immediate by successive approximations, the existence of strictly increasing singular functions lies deeper. Actually, if we except Minkowski's function $?(x),$ of which we shall speak later (and whose singularity is by no means obvious), no simple direct construction of such functions seems to be known. $[\ldots]$ Thus, it seems to be of interest to give simple direct constructions of strictly increasing singular functions."

[69] Juan Fernández Sáncheza, Pelegrí Viaderb, Jaume Paradísb, and Manuel Díaz Carrillo, A singular function with a non-zero finite derivative, Nonlinear Analysis: Theory, Methods & Applications 75 #13 (September 2012), 5010–5014.

Authors' Abstract: This paper exhibits, for the first time in the literature, a continuous strictly increasing singular function with a derivative that takes non-zero finite values at some points. For all the known "classic" singular functions--Cantor's, Hellinger's, Minkowski's, and the Riesz-Nágy one, including its generalizations and variants--the derivative, when it existed and was finite, had to be zero. As a result, there arose a strong suspicion (almost a conjecture) that this had to be the case for any singular function. We present here a singular function, constructed as a patchwork of known classic singular functions, with derivative $1$ on a subset of the Cantor set.

[70] Arthur Richard Schweitzer, An interesting class of monotonic functions, American Mathematical Monthly 16 #1 (January 1909), 4-9.

[71] H. M. Sengupta [Sen Gupta] and P. L. Ganguli, On a class of steadily increasing functions with an everywhere dense set of points of discontinuity, Bulletin of the Calcutta Mathematical Society 50 (1958), 9-18.

2nd paragraph of the paper (p. 9; italics not in original): In the present note the authors propose to link certain series of positive terms with steadily increasing functions having discontinuities at an everywhere dense set of points on $0 \leq x \leq 1.$ They use [the] notion of Harnack and a theorem due to P. Kesava Menon (1948) to achieve their objective. Theorem I (p. 10; italics in original): To each series $\sum_{k=1}^{\infty}a_{k} = 1,$ $a_{k} > 0$ and $a_k > R_{k},$ $k=1,$ $2,$ $3, \, \ldots$ we can construct a function $f(x)$ steadily increasing in $0 \leq x \leq 1$ with $f(0) = 0,$ $f(1) = 1,$ having discontinuities over the everywhere dense set of points $\{P/2^{n}\},$ where $P$ is odd, and $1 \leq P < 2^{n},$ $n=1,$ $2,$ $3, \, \ldots .$ The discontinuity at each point will be left handed or right handed according to our mode of definition.

[72] U. K. Shukla, Singular Functions and Symmetry of Derivatives, Ph.D. Dissertation (Lucknow University, India), 1954.

I have not seen this Dissertation. Garg's 1962 paper On nowhere monotone functions. II (p. 668, 1st footnote) and Garg's 1963 paper On nowhere monotone functions. III (p. 85, 2nd footnote) each state that Chapter 5 of Shukla's Dissertation gives an example of a continuous function with zero derivative almost everywhere that is monotone in no interval.

[73] Waclaw Franciszek Sierpinski, Un exemple élémentaire d'une fonction croissante qui a presque partout une dérivée nulle [An elementary example of an increasing function that has almost everywhere a zero derivative], Giornale de Matematiche di Battaglini (3) 54(7) (1916), 314-334.

Reprinted on pp. 122-140 of Sierpinski's Oeuvres Choisies, Volume II, 1975. See Denjoy (1915).

[74] Daniel Wyler Stroock, Doing analysis by tossing a coin, Mathematical Intelligencer 22 #2 (Spring 2000), 66-72.

[75] Lajos Takács, An increasing continuous singular function, American Mathematical Monthly 85 #1 (January 1978), 35-37.

For each $x \in (0,1],$ write $x = \sum_{k=0}^{\infty}2^{-n_{k}},$ where $n_{0} < n_{1} < n_{2} < \ldots$ are positive integers. For example, if $x = \frac{1}{2},$ the infinite sum $\frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \ldots$ is used. Let $b \in {\mathbb R}$ such that $b > 0$ and $b \neq 1.$ Define $f: [0,1] \rightarrow [0,1]$ by $f(0) = 0$ and, if $x \in (0,1],$ we put $f(x) = \sum_{k=0}^{\infty}b^{k}(1+b)^{-n_k}.$ Then $f$ is a strictly increasing continuous function with a zero derivative almost everywhere, with the additional property (a property that also holds for the Minkowski $?(x)$ function) that $f'(x) = 0$ at each point $x$ where $f$ has a finite two-sided derivative. This "additional property" is stated on p. 36, line 7. Takács's paper is especially notable for its literature summary at the end of the paper and for its extensive bibliography (31 items).

[76] Dale Elthon Varberg, On absolutely continuous functions, American Mathematical Monthly 72 #8 (October 1965), 831-841.

[77] Giuseppe Vitali, Analisi delle funzioni a variazione limitata, [Analysis of functions of bounded variation], Rendiconti del Circolo Matematico di Palermo (1) 46 (1922), 388-408.

[78] Donald Dines Wall, Moments of a function on the Cantor set, American Mathematical Monthly 68 #5 (May 1961), 460-461.

See Dovgoshey/Martio/Ryazanov/Vuorinen (2006, Section 5) and Evans (1957).

[79] Lui Wen, An approach to construct the singular monotone functions by using Markov chains, Taiwanese Journal of Mathematics 2 #3 (September 1998), 361-368.

Author's Abstract: "A probabilistic approach to construct the singular monotone functions by using Markov chains is given, and the relation between the singular monotone functions and the strong law of Markov chains is revealed."

[80] Adriaan Cornelis Zaanen and Wilhelmus Anthonius Josephus Luxemburg, A real function with unusual properties [Solution to Problem 5029], American Mathematical Monthly 70 #6 (June-July 1963), 674-675.

Related to results proved in Kober (1948) and Mauldon (1966).

[81] Tudor Zamfirescu, Most monotone functions are singular, American Mathematical Monthly 88 #1 (January 1981), 47-49.

[82] Tudor Zamfirescu, Typical monotone continuous functions, Archiv der Mathematik 42 #2 (1984), 151-156.