Can a function be continuous but not Hölder on a compact set? Is it possible to construct a function $f: K \to \mathbb{R}$, where $K \subset \mathbb{R}$ is compact, such that  $f$ is continuous but not Hölder continuous of any order?  It seems like there should be such a function--it would probably oscillate wildly, like the Weierstrass-Mandlebrot function.  However, the W-M function itself doesn't work, since it is Hölder.
Edit: I guess I did have in mind for the function to not be Hölder anywhere, even though I didn't explicitly say so.

No oscillation needed: the function $$f(x) = \begin{cases} 1/\log x,\quad &x\in (0,1/2], \\ 0,\quad & x=0\end{cases}$$
fits the bill. 
Indeed, $f(x)/x^\alpha\to \infty$ as $x\to0$, for any positive $\alpha$.
