Bibliography for Singular Functions [closed]

[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.