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Takagi function revisited

Sadowski Ł.

Journal of Applied Analysis

2017

Vol. 23, nr 2

85--88

Starting with some equations that characterize the Takagi function in the class of bounded functions, we come naturally to a family of functional equations and extend two classical results by H.-H. Kairies to that family.

Paraconvex, but not strongly, Takagi functions

Tabor J., Tabor J.

Control and Cybernetics

2012

Vol. 41, no. 3

545-559

There is an important open problem in the theory of approximate convexity whether every paraconvex function on a bounded interval is strongly paraconvex. Our aim is to show that this is not the case. To do this we need the following generalization of Takagi function. For a sequence a = (ai)i∈N ⊂ R+ we consider Takagi-like function of the form T(a)(x) := ∑ ∞ i=1 aidist(x, 12i-1Z) for x ∈ R. We give convenient conditions for verification whether T(a) is paraconvex or strongly paraconvex. This enables us to construct a class of paraconvex functions which are not strongly paraconvex.