Doubly paradoxical functions of one variable

• Published in: Journal of mathematical analysis and applications 2018-08, Vol.464 (1), p.274-279
• Main Authors: Ciesielski, Krzysztof C., Pan, Cheng-Han
• Format: Article
• Language: English
• Subjects: Continuous nowhere differentiable; Differentiable monster; Differentiable nowhere monotone; Jarník's Extension Theorem; Pointwise shrinking globally stable map; Weierstrass's monster
• ISSN: 0022-247X; 1096-0813
• DOI: 10.1016/j.jmaa.2018.04.012

This paper concerns three kinds of seemingly paradoxical real valued functions of one variable. The first two, defined on R, are the celebrated continuous nowhere differentiable functions, known as Weierstrass's monsters, and everywhere differentiable nowhere monotone functions—simultaneously smooth and very rugged—to which we will refer as differentiable monsters. The third kind was discovered only recently and consists of differentiable functions f defined on a compact perfect subset X of R which has derivative equal zero on its entire domain, making it everywhere pointwise contractive, while, counterintuitively, f maps Xonto itself. The goal of this note is to show that this pointwise shrinking globally stable map f can be extended to functions f,g:R→R which are differentiable and Weierstrass's monsters, respectively. Thus, we pack three paradoxical examples into two functions. The construction of f is based on the following variant of Jarník's Extension Theorem: For every differentiable function f from a closedP⊆RintoRthere exists its differentiable extensionfˆ:R→Rsuch thatfˆis nowhere monotone onR∖P.