Smallest positive zero of Weierstrass nowhere differentiable function

Question

Consider the Weierstrass nowhere differentiable function $f(x) = \sum_{n=0}^\infty \frac{1}{2^n} \cos(4^n \pi x)$. It seems that the smallest positive zero of $f(x)$ occurs at $x=\frac{1}{5}$, but I haven't been able to prove this. Can anyone prove or disprove this statment?

Answer 1

I am using your question to give detailed summaries and comments on some little known work involving zeros and level sets of continuous nowhere differentiable functions done by Indian mathematicians from about 1930 to about 1950. For the most part the work itself is not especially profound, and often the work was not well informed with what was then generally known by other researchers in real variable theory. However, the questions addressed are interesting and they do not seem to have been considered elsewhere in the literature during this period. Also, it is my opinion that a survey of this work, from the initial papers to the more sophisticated later papers, makes for an excellent expository vehicle to understand the far more technical work done by Jack B. Brown, Andrew M. Bruckner, K. M. Garg, Marius Iosifescu, Solomon Marcus, Simion Stoilow, and others.

Conventions: Direct quotes from original sources are indicated by italics. Within such a quote: (1) italics in the original is indicated by non-italics here; (2) brief additions and/or words of further explanation by me are indicated by non-italics between square braces "[stuff by me]"; (3) omissions in the original are indicated using "$[\dots]$". A bold-face Note indicates a note by me that is longer, typically when one or more sentences are used. In some of the cases where I have stated theorems separately, the original theorem statements were italicized and I have italicized them here as well (thus deviating from my default policy). I have tried very carefully to preserve the original punctuation and spelling (some sentences ending in mathematical expressions did not end in periods in the original), but I have not included any indications of where paragraphs begin and end in the original.

Glossary of older terms: enumerable means countable; unenumerable means uncountable; progressive derivative means right derivative; regressive derivative means left derivative; non-dense means nowhere dense; everywhere dense means dense in the smallest interval containing the set (sometimes "dense", without being prefaced by the "everywhere", means in context "everywhere dense"); incrementary ratio means the ordinary difference quotient that a derivative is the limit of; line of invariability (of a function) is an interval on which the function is constant; the set exists means the set is not empty.

Warning: In many of the excerpts "non-differentiable function" is used without the use of "continuous". However, in all cases where "non-differentiable function" appears in the excerpts, the meaning is "nowhere differentiable continuous function" unless explicitly stated otherwise (by me or by the original author). I'm fairly certain it was the convention of these Indian mathematicians that "non-differentiable" usually meant "nowhere finitely or infinitely differentiable", which would exclude the Takagi function, for instance, since the Takagi function has a two-sided derivative of $+\infty$ on a certain countable dense set and a two-sided derivative of $-\infty$ on another such set. However, in almost all cases the distinction is not relevant. I have tried to explicitly state what "non-differentiable" means in a particular paper when I thought the meaning was important.

[1] Ganesh Prasad (1876-1935), On the zeroes of Weierstrass's non-differentiable function, Proceedings of the Benares Mathematical Society 11 (1929), 1-8. JFM 55.0144.02

Besides this paper, Prasad published papers on the potential of ellipsoids with variable densities, the conduction of heat, expansions of functions by spherical harmonics, the mean value theorem for derivatives, summation of infinite series of Legendre's functions, and other topics. He wrote several books, including a book on elliptic functions, a book on spherical harmonics, a historical book on mathematical physics and differential equations at the beginning of the 20th century, and Some Great Mathematicians of the 19th Century (2 volumes).

The following excerpt from p. 53 of Prasad's biographical sketch of Ernest William Hobson in [Bulletin of the Calcutta Mathematical Society 25 (1933), pp. 31-54] may be of interest: As the very first course of lectures ever delivered at Cambridge on the theory of functions of a real variable was the course delivered by Hobson in 1902, it may be concluded that Hobson became devoted to the study of that subject about 1900. The course referred to above was attended by Professors H. F. Baker [Henry Frederick Baker, 1866-1956] and G. B. Mathews [George Ballard Mathews, 1861-1922], and three senior students including myself. [Prasad had a Ph.D. at this time, so I suspect "senior students" describes those students who already had a Ph.D. (or terminal) degree.]

For biographies of Prasad, see: (1) S. C. Bagchi, "In Memoriam. Dr. Ganesh Prasad. 1876-1935, Bulletin of the Calcutta Mathematical Society 27 #1-2 (1935), 93-98 [Correction in BCMS 28 (1936), pp. 121-122]; (2) Prof. Ganesh Prasad, Nature 135 #3417 (27 April 1935), 644.

First 3 sentences of the paper: The object of the present paper is chiefly to give a general expression from which the zeroes of Weierstrass's non-differentiable function $W(x) \equiv \sum_{0}^{\infty} \frac{cos (x \pi \cdot {13}^{n})}{2^n}$ can be calculated. It is easy to see that a similar expression can be investigated in the case of the general function $w(x) \equiv \sum_{0}^{\infty} a^{n} \cos(x\pi \cdot b^{n}),$ $b$ being an odd integer and $a$ such a proper fraction that $ab > 1 + \frac{3\pi}{2};$ such an expression has also been given in this paper. The expressions given by me are believed to be new.

From the bottom of p. 2: The zeroes of $W(x)$ between $0$ and $1$ are $\frac{1}{2}$ and $\frac{1}{2} \pm \frac{1}{{13}^{k}}(1 + \frac{1}{2}\lambda_{k}),$ where $k$ is any positive integer and $\lambda_k$ is a fraction between $\frac{4}{13}$ and $\frac{6}{13}$ which can be approximated to as closely as we please.

NOTE: I believe "fraction" here simply means a real number between $0$ and $1.$ Also, the p. 2 sentence should be modified to begin as "Among the zeros of $W(x)$ between $0$ and $1$ are $\ldots$" There are in fact continuum many zeros, and Prasad's subsequent argument only shows the existence of those zeros that he states to be zeros. [Prasad's argument is by making suitable estimates of the locations of intervals within which the values of $W(x)$ change sign.] Prasad discusses the issue of all the zeros on p. 5.

From the bottom of p. 4: (3 footnotes omitted) Although no previous writer seems to have made any definite attempt to obtain the zeroes of $W(x)$ or $w(x);$ those, who, like C. Wiener, F. Klein and G. C. Young, attempted a "graphical representation," must have desired to know the zeroes.

From p. 5: After the preceding articles had been printed off, it came to my mind that, in my desire to stress that an infinite set of zeroes of $W(x)$ with $\frac{1}{2}$ as its limiting point can be actually found, I was slightly inaccurate in using the definite article "the" before the word "zeroes" in the second line of p. 1 and in the first line of Article 2 and giving thereby the impression to the reader that according to me no other zeroes of $W(x)$ existed in the interval $(0,1).$ [Note: The first 4 pages of the paper consist of the first 3 sentences I gave above followed by articles [= sections] numbered 1 through 4. The present excerpt is the beginning of Article 5.] I proceed now to prove the following statement to be referred to later as $(u):$ There is a zero of $W(x)$ between $\frac{1}{2} + \frac{3/2}{{13}^{k}}$ and $\frac{1}{2} + \frac{2}{{13}^{k}},$ another between $\frac{1}{2} + \frac{3}{{13}^{k}}$ and $\frac{1}{2} + \frac{7/2}{{13}^{k}},$ a third between $\frac{1}{2} + \frac{7/2}{{13}^{k}}$ and $\frac{1}{2} + \frac{4}{{13}^{k}},$ a fourth between $\frac{1}{2} + \frac{5}{{13}^{k}}$ and $\frac{1}{2} + \frac{11/2}{{13}^{k}},$ and lastly a fifth between $\frac{1}{2} + \frac{11/2}{{13}^{k}}$ and $\frac{1}{2} + \frac{6}{{13}^{k}};$ further, the first three of these zeroes, at least for $k=1,$ and the last two zeroes are other than those given by the expression investigated in Art. 2; also there are zeroes in the interval $(0, \, \frac{1}{2})$ which are situated symmetrically to the five given above for $(\frac{1}{2}, \, 1).$ [Note: In the original the last four appearances of ${13}^k$ were $13$ (no exponent of $k$), but these were almost certainly typos.]

The 7th and final section of the paper, on p. 8: In concluding this paper, I may add that I was led to investigate the zeroes of Weierstrass's non-differentiable function in connection with my recent researches relating to the properties of Rolle's function $\theta$ in the mean-value theorem $f(x+h)=f(x)+hf'(x+\theta h).$ That the subject of the zeroes of non-differentiable functions is of great interest by itself, needs no emphasis. Two problems which suggest themselves to me in connection with this subject may be taken up by me for study in a subsequent paper; the problems are: (i) To prove rigorously that there are no zeroes in the interval $(0,1)$ in the case of $W(x)$ or $w(x)$ other than those given in this paper. (ii) To classify non-differentiable functions according to the number of the limiting points which its zeroes possess in a finite interval; it being of course understood that two functions are for the purpose of this classification to be considered of the same class if they differ by a finite rational or trigonometrical polynomial.

[2] Jhamman Lal Sharma (??-??), On the zeros of Weierstrass's non-differentiable function, Proceedings of the Benares Mathematical Society 11 (1929), 11-22. JFM 55.0144.03

First 4 sentences of the paper: The object of the present paper is to give the zeros of Weierstrass's nondifferentiable function $W(x) \equiv \sum_{0}^{\infty}\frac{x \, \pi \, {13}^{n}}{2^n}$ more approximately than has been done by Professor Ganesh Prasad in a paper recently published by him under the same title as the present paper. Professor Prasad proves that [some of] the zeros in the interval $(0,1)$ are given by the expression $\frac{1}{2} + \frac{1}{{13}^{k}}\left(1+ \frac{1}{2} \lambda_{k}\right)$ where $k$ is any positive integer and $\lambda_k$ lies between $\frac{4}{13}$ and $\frac{6}{13}$ Prof. Prasad has stated that $\lambda_k$ can be approximated to as closely as we please; this investigation is undertaken by me at his suggestion. I obtain the value of $\lambda_k$ correct to $5$ digits in the scale of $13,$ and of the root correctly upto [sic] six digits.

From p. 20 (lines 10 to 11): Therefore $\lambda_k$ when $k > 1$ lies between $\frac{4}{13} + \frac{3}{{13}^2} + \frac{9}{{13}^3} + \frac{3}{{13}^4} + \frac{11}{{13}^5}$ and $\frac{4}{13} + \frac{3}{{13}^2} + \frac{9}{{13}^3} + \frac{4}{{13}^4}$

From p. 22 (line -4): Therefore $\lambda_1$ lies between $\frac{4}{13} + \frac{3}{{13}^3} + \frac{9}{{13}^4}$ and $\frac{4}{13} + \frac{3}{{13}^3} + \frac{9}{{13}^4} + \frac{2}{{13}^5}$

[3] Santosh Kumar Bhar (??-??), On the zeros of non-differentiable functions of Darboux's type, Bulletin of the Calcutta Mathematical Society 22 #2-3 (1930), 61-90. JFM 56.0237.01

This paper was motivated by a recent investigation into the zeros of the Weierstrass function by Prasad (1929). Bhar, at the suggestion of Prasad (presumably in person), investigates some zeros of functions given by Darboux in 1879 and some zeros of functions belonging to a more inclusive class of functions given by Lerch in 1888. In a footnote on p. 61 Bhar points out that these functions are nowhere finitely differentiable, but they do have points at which an infinite derivative exists. I suspect each of the functions has continuum many zeros, but I do not believe Bhar mentions anywhere the possibility that any of the functions has even uncountably many zeros. Most of Bhar's results involve the location of zeros in various intervals whose endpoints are given by explicit factorial-like expressions. However, in at least one place, the first sentence on p. 63, Bhar states (probably incorrectly) that a certain countable set of points gives all the zeros of one of the functions.

From pp. 61-62: (3 footnotes omitted) It is believed that those, who wished to give "graphical representations" of non-differentiable functions, as Wiener, Felix Klein and G. C. Young did in the case of Weierstrass's function, must have desired to know the zeros of those functions. However, it is only recently that the first successful investigation of this kind has been published by Professor Ganesh Prasad who has given general expressions from which zeros of Weierstrass's function $\sum_{0}^{\infty} a^n \cos \left(b^n \pi x\right)$ can be obtained. Professor Prasad has also suggested the following problem: "To classify non-differentiable functions according to the number of limiting points which its zeros possess in a finite interval."

From p. 89: Hitherto, our attention has been mainly directed to search out one single limiting point of zeros of non-differentiable functions. We will conclude this paper, with an investigation of some other limiting points, which are, however, finite in number, in the interval $(0,1).$

[4] Avadhesh [Avaresh] Narayan Singh (1901-1954), On the unenumerable zeros of Singh's non-differentiable function, Bulletin of the Calcutta Mathematical Society 22 #2-3 (1930), 91-98. JFM 56.0237.02

Singh obtained his Ph.D. (under Ganesh Prasad) in 1929 (1928?) for his work on the arithmetically defined continuous nowhere differentiable functions that appear in this paper. His research papers consist of 24 papers on the differentiability properties of real valued functions of one real variable, 3 papers on space-filling and fractal curves (Peano curve, Koch curve, etc.), 2 papers involving pointwise divergence of Fourier series, 14 papers on the history of mathematics, and a paper (his first, published around 1924) on the evaluation of certain definite integrals. His books include a 1935 monograph on continuous nowhere differentiable functions published in 1935 (listed as [7] below) and two books co-authored with Birhy Bibhutibhusan [Bibhuti Bhusan] Datta on the history of Hindu mathematics.

For a biography of Singh, see: Shambhu Dayal Sinvhal, Dr. Avadhesh Narain [sic?] Singh (a life sketch), Ganita (Lucknow University) 5 (1954), i-vii.

Introduction (p. 91): (2 footnotes omitted) In a paper recently published in the Proceedings of the Benares Mathematical Society, Vol XI, Prof. G. Prasad has located zeros of Weierstrass's non-differentiable function. The zeros that have been found out by him form an enumerable set with the point $x = \frac{1}{2},$ as a limiting point. The present investigation was suggested by Prof. Prasad's work. The object of this paper is to study the zeros of a class of Non-differentiable functions defined by me in a paper published in the Annals of Mathematics, Vol. 28, (1927), pp. 472-76. As these functions are arithmetically defined, the location of the zeros presents little difficulty. It has been shown that the set of the zeros of any of these functions $\theta(t)$ possesses the following properties: (a) it is unenumerable; (b) it is perfect; (c) it has zero measure. It has been further pointed out that the roots of the equation $\theta(t) = c,$ $0<c<1,$ form a set similar to the set of the zeros.

Last paragraph of the paper: (1 footnote omitted) In my paper "On Infinite Derivates" published in this Bulletin, Vol. 16, pp. 79, I have given an example of a continuous function $f(x)$ whose zeros form a set of positive measure. The function $f(x)$ is, however, not non-differentiable. The method of construction employed by me can be easily modified to give an example of a non-differentiable function $f(x)$ whose zeros form a set of positive measure. Thus the zeros of a continuous non-differentiable function may form a set of positive measure. [Note: In a footnote Singh points out that Volterra's function (a function whose derivative exists everywhere and is uniformly bounded, but whose derivative is not Riemann integrable) also has a zero set with positive measure. Incidentally, in this paper by Singh "non-differentiable" means "continuous and having at no point a finite or infinite two-sided derivative".]

Note: The correct title and full reference for the paper Singh cites as "On Infinite Derivates" published in this Bulletin, Vol. 16, pp. 79 is On the derivates of a function, Bulletin of the Calcutta Mathematical Society 16 #2 (1925), 79-88. The example that Singh cites in the last paragraph of his 1930 paper appears to be correct. In particular, Singh's example does not have the error discussed at the beginning of Lipinski's 1966 paper On zeros of a continuous nowhere differentiable function. The example, which is given on pp. 87-88 of Singh's 1925 paper, is the function $f$ defined to be $0$ on the symmetric Cantor set $C\left\{\frac{1}{3^n}\right\}$ (at the $n$'th construction stage divide each of the $2^{n-1}$ many closed intervals into $3^n$ many abutting open subintervals of equal length and remove the middle open subinterval) and, on each of the bounded complementary open intervals $(a,b),$ the graph of $f$ is an isosceles triangle whose base is that open interval and whose height is $\sqrt{b-a}.$ Singh shows that at each $\xi \in C\left\{\frac{1}{3^n}\right\}$ such that $\xi$ is not an endpoint of a complementary interval of $C\left\{\frac{1}{3^n}\right\}$ (i.e. $\xi$ is a bilateral condensation point of $C\left\{\frac{1}{3^n}\right\},$ which includes all but a countable subset of $C\left\{\frac{1}{3^n}\right\})$ we have $D_{-}f(\xi)=-\infty$ and $D^{-}f(\xi)=0$ and $D_{+}f(\xi)=0$ and $D^{+}f(\xi)=+\infty$. The modification Singh probably has in mind is to replace the triangles over the complementary intervals with nowhere differentiable continuous functions that are zero at the endpoints of the interval, positive at each point in the interior of the interval, and whose maximum heights over the intervals are the same as that of the triangles.

[5] Bholanath Mukhopadhyay [Mookerji] (??-??), On the limiting points of the zeros of a non-differentiable function first given by Dini, Bulletin of the Calcutta Mathematical Society 22 #2-3 (1930), 103-114. JFM 56.0237.03

First two paragraphs of the paper: (1 footnote omitted) The publication of Prof. G. Prasad's remarkable paper on the zeros of Weierstrass's non-differentiable function has naturally led mathematicians to study the zeros of different types of non-differentiable functions. The object of the present paper is chiefly to ascertain whether in a finite interval, say, $(0,1)$ there is any limiting point of the zeros of the non-differentiable function $F(x) = \sum_{1}^{\infty} \frac{\sin ({16}^{n} \pi x)}{2^n}$ and as also of the general non-differentiable function $f(x) = \sum_{1}^{\infty} a^n \sin b^n \pi x \;$ where $\;0<a<1$ and $b$ is an even integer and $\;ab > 1 + \frac{3\pi}{2}.$ [Note: The results stated on p. 109 together with the results stated on p. 111 imply that each of the $17$ points $x = \frac{k}{16}$ $\;(k=1,\,2,\,\ldots, \, 17)\;$ is a limit point of the the zero set of $F(x).$]

[6] Avadhesh [Avaresh] Narayan Singh (1901-1954), On the method of construction and some properties of a class of non-differentiable functions, Proceedings of the Benares Mathematical Society 13 (1931), 1-17. Zbl 7.15306; JFM 57.0303.05

Section 1 of the paper (pp. 1-2): (4 footnotes omitted) In the year 1890 Peano defined arithmetically two functions $\phi(t)$ and $\psi(t)$ which he stated without proof to be nondifferentiable functions of $t.$ A geometrical construction for a class of functions, of which Peano's functions are special cases, has been given by Moore who has proved with the help of his geometrical definition that these functions are non-differentiable. It is believed that no analytical proof of the non-differentiability of Moore's functions has been given by any previous writer. In this paper I give a general class of arithmetically defined functions, of which Moore's and Peano's functions are special cases. The functions are studied purely with reference to their arithmetic definition, and a direct analytical proof of their non-differentiability is given. These functions can also be obtained by a method given by E. Steinitz. The advantage of the arithmetic definition is the ease with which we get full information as regards the values of the [Dini] derivates at the various points. Besides proving the non-differentiability of the functions, I have also found out the points where the functions possess progressive and regressive derivatives [= right and left derivatives] an information which Steinitz's analysis fails to supply regarding the functions considered by him. It is well known that a non-differentiable function is everywhere oscillating. Our method of definition enables us to study in detail the character of these oscillations in the case of the functions given here, for instance, it can be easily shown that at no point do any of these functions possess a cusp, and that every value of the function is attained at a perfect set of points, i.e., the functions do not possess proper maxima or minima. [Note: Regarding the matter of cusps (see Srinivasiengar's paper below for the definition), in a footnote on p. 8 Singh quotes the following from Hobson's The Theory Of Functions Of A Real Variable (Volume 2, 1926, 2nd edition, p. 405, last sentence): It does not appear to be definitely known whether a non-differentiable function can exist which has no cusps.]

From the bottom of p. 2: [This pertains to the definition of Singh's continuous nowhere differentiable functions $\phi_{m,r,p}$.] In the above, the integer $p$ may have any odd value $(3,\,5,\,7,\,\ldots,)$ the integer $m$ may have any one of the values $2,\,3,\,4,\,\ldots,$ and $r$ is an integer $\leq m.$ The function $\phi_{2,1,3}(x)$ is Peano's function $\phi(t)$ $[= x$-coordinate function of Peano's space filling curve], and the function $\phi_{2,2,3}(x)$ is Peano's function $\psi(t)$ $[= y$-coordinate function of Peano's space filling curve]; while the functions $\phi_{2,1,p}(x)$ and $\phi_{2,2,p}(x),$ where $p$ is an odd integer $\geq 3,$ are the functions considered by Moore [in his 1900 paper On certain crinkly curves].

Section 8 (pp. 14-17; last section of the paper): Singh shows that the solutions to $\phi_{3,1,3}(x) = c$ in the interval $[0,1]$ form a nonempty perfect set of measure zero, where $c$ is any fixed real number such that $0 \leq c \leq 1.$ Singh remarks (p. 14, lines -7 to -6) that the same properties can be proved in a similar way for any of the functions $\phi_{m,r,p}.$ This last section of Singh's paper is, as far as I can tell, word-for-word of what can be found on pp. 89-93 of Singh's 1935 monograph.

[7] Avadhesh [Avaresh] Narayan Singh (1901-1954), The Theory and Construction of Non-Differentiable Functions, Lucknow University Studies #1, Newul Kishore Press [Lucknow University Press], 1935, vii + 110 pages. JFM 61.1113.06 [Reprinted in Squaring the Circle and Other Monographs, Chelsea Publishing Company, 1953 (MR 14,1114a; Zbl 52.16301).]

Reviews of the 1935 monograph: (1) Frédéric Amédée Emile Roger, Bulletin des Sciences Mathématiques (2) 60 (1936), 323 (in French); (2) Carl Einar Hille, Bulletin of the American Mathematical Society 45 #3 (March 1939), 217-218. Reviews of the 1953 collection: (3) Ernest Albert Hedberg, The Pentagon 14 #1 (Fall 1954), 50-51; (4) Thomas Arthur Alan Broadbent, Mathematical Gazette 39 #327 (February 1955), 87.

From pp. 87-89: (2 footnotes omitted) It is evident that Weierstrass's function $W(x) = \sum a^n \cos(b^n \pi x)$ (where $b$ is an odd integer) is zero at $x = \frac{1}{2}.$ At this point, the derivates on the right as well as on the left oscillate between $\infty$ and $-\infty.$ It follows that the point $x = \frac{1}{2}$ is a limiting point of the zeros of the function $W(x).$ A set of zeros of $W(x)$ with $x = \frac{1}{2}$ as a limiting point has been actually located by G. Prasad. His result in the case of the function $W(x) = \sum \frac{\cos \left( {13}^{n} \pi x \right)}{2^n}$ may be stated as follows: There is a zero of $W(x)$ between (i) $\left(\frac{1}{2} \pm \frac{1}{{13}^k}\right)$ and $\left(\frac{1}{2} \pm \frac{3/2}{{13}^k}\right);$ (ii) another between $\left(\frac{1}{2} \pm \frac{3/2}{{13}^k}\right)$ and $\left(\frac{1}{2} \pm \frac{2}{{13}^k}\right);$ (iii) a third between $\left(\frac{1}{2} \pm \frac{3}{{13}^k}\right)$ and $\left(\frac{1}{2} \pm \frac{7/2}{{13}^k}\right);$ (iv) a fourth between $\left(\frac{1}{2} \pm \frac{7/2}{{13}^k}\right)$ and $\left(\frac{1}{2} \pm \frac{4}{{13}^k}\right);$ (v) a fifth between $\left(\frac{1}{2} \pm \frac{5}{{13}^k}\right)$ and $\left(\frac{1}{2} \pm \frac{11/2}{{13}^k}\right);$ (vi) and a sixth between $\left(\frac{1}{2} \pm \frac{11/2}{{13}^k}\right)$ and $\left(\frac{1}{2} \pm \frac{6}{{13}^k}\right).$ The above is not a complete list of all the zeros of the function $W(x).$ The determination of all the zeros is probably not practically possible. Bhar has given a list of some of the zeros of the function $\sum \frac{ {16}^{n} \cos \{1.\,3.\,5.\, ... \, (2n-1)\} \pi x}{1.\,3.\, 5. \, ... \, (2n-2)}$ [Note: In place of "$\cos$", the original indicates either "$\cos$" or $\sin$" can appear.] Prasad's result quoted above is a verification of the conclusion that we can arrive at by a consideration of the values of the derivates at the point $x = \frac{1}{2}.$ It would be interesting to find some special character of the set of zeros, e.g., whether they form an enumerable set or not, or whether the set is closed or open. Very interesting results regarding the nature of the oscillations of a non-differentiable function have been recently obtained by studying the intersections of the line $y=c$ with the curve $y = \phi_{m,r,p}(x)$ given in the third lecture. It has been found that the roots of the equation $\phi_{m,r,p}(x) = c$ $(0 \leq c \leq 1),$ form a set $S_c$ which is perfect and is of zero measure. Thus the oscillations of the function $\phi_{m,r,p}(x)$ in every interval, ever-so-small, are unenumerable. The function is thus much more complicated than ordinary transcendental functions which may have an enumerably infinite number of oscillations in an interval. It may be pointed out that by the help of such a function we can easily express the continuum in $(0,1)$ as an unenumerable aggregate of unenumerable aggregates.

[8] C. N. Srinivasiengar (1900-1972), On the zeros of Weierstrass's non-differentiable function, Journal of the Indian Mathematical Society (N.S.) 3 #3 (September 1938), 114-117. Zbl 19.40102; JFM 64.0205.02

The paper begins with a discussion of Prasad's results concerning zeros of the Weierstrass function (for $a = \frac{1}{2}$ and $b = 13).$ Srinivasiengar gives the approximate values that Prasad found of the terms in several sequences of zeros that approach the zero $x = \frac{1}{2}$ on the left, as well as the approximate values that Prasad found of the terms in several sequences of zeros that approach the zero $x = \frac{1}{2}$ on the right.

From pp. 114-115: (2 footnotes omitted) On the other hand, A. N. Singh has shown that for his function (an extended form of [the coordinate functions of] Peano's [space-filling] function), every zero is a limiting point of the zeros. That such a property is nearly true for Weierstrass's function will be pointed out in this paper. Ganesh Prasad evidently did not consider this possibility. Singh also writes, "It would be interesting to find some special character of the set of zeros, e.g. whether they form an enumerable set or not, or whether the set is closed or open."

On p. 115 Srinivasiengar defines knot point and cusp of a function $f.$ A knot point of $f(x)$ is a point $x=a$ such that $D_{-}f(a) = D_{+}f(a) = -\infty$ and $D^{-}f(a) = D^{+}f(a) = +\infty.$ The term originates from p. 168 of Grace C. Young's 1916 essay On infinite derivates, and in this essay she proved that for the Weierstrass function almost all points in both the measure sense and the Baire category sense are knot points (i.e. the set of points that are not knot points of the Weierstrass function has Lebesgue measure zero and is first category). A knot point can be thought of as the most extreme way that a derivative can fail to exist at a point, at least if we exclude the numerous generalized and/or technical notions that I will not deal with here (e.g. approximate and other forms of generalized Dini derivates, or when the difference quotient limits involve division by $\sqrt{h}$ and the like instead of division by $h).$

THIS HISTORICAL/LITERATURE SURVEY CONTINUES. SEE MY ANSWER AT

Level sets of a Weierstrass nowhere-differentiable function