Can anyone describe level sets of a Weierstrass nowhere-differentiable function? For example, let $f(x) = \sum_{n=0}^\infty \frac{1}{2^n} \cos( 4^n \pi x)$. For some $c \in (-2,2)$, what is known about the set $f^{-1}(c)$? How about the sublevel set $f^{-1}(-\infty, c]$?
[Math] Level sets of a Weierstrass nowhere-differentiable function
fractalsreal-analysis
Best Answer
This is a continuation of my answer to Smallest positive zero of Weierstrass nowhere differentiable function
[9] Ramaswamy Vaidyanathaswamy (1894-1960), On continuous functions of a real variable, Proceedings of the Indian Academy of Sciences. Section A 9 #1 (January 1939), 67-71. Zbl 21.01502; JFM 65.0195.03
[10] Subbaramiah [Subramiah] Minakshisundaram (1913-1968), On the roots of a continuous non-differentiable function, Journal of the Indian Mathematical Society (N.S.) 4 #1 (March 1940), 31-33. MR 1,303a; Zbl 23.02001; JFM 66.1224.02
[11] Zygmunt Zahorski (1914-1998), Sur l'ensemble des racines de l'équation $W(x) = f(x)$ [On the set of roots of the equation $W(x) = f(x)],$ Comptes Rendus des Séances de la Société des Sciences et des Lettres de Varsovie [Varsovie = Warsaw]. Classe III 41 (1948), 43-45. MR 13,219a; Zbl 39.05604
[12] K. Padmavally (??-??), On the roots of equation $f(x) = \xi$ where $f(x)$ is real and continuous in $(a,b)$ but monotonic in no subinterval of $(a,b)$, Proceedings of the American Mathematical Society 4 #6 (December 1953), 839-841. MR 15,513h; Zbl 52.05201
Character posting limitations (30,000 characters, including spaces, per answer) prevent me from saying anything about the following two papers. However, in my opinion these two papers best follow those that I have already discussed, both from a mathematical perspective and from a historical perspective.
[13] Solomon Marcus (1925- ), Sur les fonctions continues qui ne sont monotones en aucun intervalle [On continuous functions that are not monotone in any interval], Académie de la République Populaire Roumaine. Revue de Mathématiques Pures et Appliquées [after 1963: Revue Roumaine de Mathématiques Pures et Appliquées] 3 (1958), 101-105. MR 21 #6403; Zbl 87.27402
[14] Krishna Murari Garg (1932- ), On level sets of a continuous nowhere monotone function, Fundamenta Mathematicae 52 (1963), 59-68. MR 26 #1405; Zbl 106.26904