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I want to prove the existence of a $f:C([0,1])\to \mathbb{R}$ such that $$\lim_{h\to0}\sup|h|^{-\alpha}|f(x+h)-f(x)|\to \infty$$ for all $x\in [0,1]$. Where $\alpha > 0$.

I was given the hint that I should consider the sets $E_m$ consisting of those $f ∈ C([0, 1])$ for which there exists an $x ∈ [0, 1]$ with $|f(x + h) − f(x)| \le m|h|^ α$ for all $x + h ∈ [0, 1]$. And then applying some version of Baire category. I am really flummoxed by this however as since we only know what these $f$ do at a single $x\in [0,1]$ it is really difficult to see how we can combine this to get a result that holds at all $x\in [0,1]$.

Elie Bergman
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Outline: First show that every $E_m$ is closed and nowhere dense. As $C([0,1])$ is a complete metric space, the Baire category theorem implies that there must be $f \in C([0,1])$ such that $f \notin E_m$ for all $m \in \mathbb{N}$.

user159517
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