Monsters in Calculus

One of the strangest, most mind-boggling examples in analysis is that of a function from to that is everywhere differentiable but monotone on no interval. The graph of such a "monstrous" function is simultaneously smooth and very rugged. Although such examples have been known for over 100...

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Bibliographic Details
Published in:The American mathematical monthly 2018-09, Vol.125 (8), p.739-744
Main Author: Ciesielski, Krzysztof Chris
Format: Article
Language:English
Subjects:
Calculus
Mathematical functions
MSC: Primary 26A24
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Summary:One of the strangest, most mind-boggling examples in analysis is that of a function from to that is everywhere differentiable but monotone on no interval. The graph of such a "monstrous" function is simultaneously smooth and very rugged. Although such examples have been known for over 100 years, so far the existing constructions are quite involved. In this note, we provide a simple example of such a map. It is presented in a broader context of other paradoxical examples related to differentiability of continuous maps from to , including a differentiable function that maps a compact perfect subset of onto itself even though its derivative vanishes on .
ISSN:0002-9890
1930-0972
DOI:10.1080/00029890.2018.1502011