> [!theorem] > > Let $\alpha > 0$, then there exists a [[Continuity|continuous]] function $f \in C([0, 1], \real)$ such that $f$ is nowhere locally [[Hölder Space|Hölder]] $\alpha$-continuous. > > *Proof*. For each $h \le 1$ and $f \in X$, denote > $ > \Delta_h(f) = \inf_{x \in [h, 1 - h]}\max\bracs{\frac{\abs{f(x + h) - f(x)}}{h^\alpha}, \frac{\abs{f(x - h) - f(x)}}{h^\alpha}} > $ > then $\Delta_h(f): C([0, 1]) \to \real$ is continuous with respect to the [[Uniform Norm|uniform norm]]. For each $N \in \nat$, let > $ > A_{h, N} = \bracs{f \in C([0, 1]): \Delta_h(f) > N} \quad U_N = \bigcup_{n \in \nat}A_{1/n, N} > $ > Let $f \in A_{1/n, N} \subset U_N$, then for each $g \in C([0, 1])$, > $ > \begin{align*} > \frac{\abs{g(x + h) - g(x)}}{h^\alpha} &\ge \frac{\abs{f(x + h) - f(x)} - 2\norm{f - g}_u}{h^\alpha} > \end{align*} > $ > so $U_N$ is open. > > On the other hand, let $f \in C([0, 1])$ and $\eps > 0$. Let $F \in C([0, 1])$ be a piecewise linear function with slope greater than $N$ and $\norm{F - f}_u < \eps$, then $F \in U_N$. Therefore $U_N$ is dense in $C([0, 1])$. > > By the [[Baire Space|Baire category theorem]], $\bigcap_{N \in \nat}U_N$ is dense and hence non-empty, where any element of $U_N$ is nowhere locally Hölder-$\alpha$ continuous.