As per the title. I know that a Lipschitz continuous function is differentiable almost everywhere (see the Rademacher Theorem). I was wondering if something similar was true for Hölder continuous functions.

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    $\begingroup$ For each $s < 1,$ there are nowhere differentiable continuous functions that are Hölder continuous with Hölder exponent $s.$ I'm pretty sure this was known by the 1890s (at least), but off-hand I don't know any really old specific references. A very strong result is Theorem 2.1 on p. 45 of On continuous functions with no unilateral derivatives by Masayoshi Hata (1988), which is freely available on the internet. $\endgroup$ Apr 12, 2021 at 14:18

1 Answer 1


The Weierstrass function is Hölder continuous but differentiable nowhere.


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