An inequality for the Takagi function

Abstract

(where, throughout this paper, R, Z, and N denote the sets of real numbers, integers, and positive integers, respectively, N0 = N∪{0}, and dist(x,Z) = inf{ |x−s| : s ∈ Z }). Functions of this type have been investigated by several authors (e.g. [3], [7], and [10]) as convenient examples for continuous nowhere differentiable functions. In particular, function T is usually cited as “van der Waerden’s function” (e.g. [1], [2]). However, as it was also mentioned by Knopp [7], function T had been constructed earlier by Takagi [9] on the interval [0, 1] in a somewhat different way. Namely, Takagi determined T (x) with the aid of the dyadic expansion of x. It seems therefore historically correct to call T the Takagi function. More historical and mathematical details can be found, for instance, in Kairies’ paper [6]. Recently, Hazy and Pales have discovered that the Takagi function plays a specific role in the theory of approximately convex functions. Namely, in order to extend the celebrated theorem of Bernstein and Doetsch for approximately midconvex functions, they have proved the following result [4, Theorem 4]. Theorem A. Let X be a normed linear space and let D ⊂ X be an open convex set. Suppose that e and δ are nonnegative real numbers, f : D → R fulfils the inequality