Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
8899581 | Journal of Mathematical Analysis and Applications | 2018 | 6 Pages |
Abstract
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:RâR 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 JarnÃk's Extension Theorem: For every differentiable function f from a closedPâRintoRthere exists its differentiable extensionfË:RâRsuch thatfËis nowhere monotone onRâP.
Related Topics
Physical Sciences and Engineering
Mathematics
Analysis
Authors
Krzysztof C. Ciesielski, Cheng-Han Pan,