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.

Take a look at the functon
$$f(x) = \sum_{n=0}^\infty {\sin(2^n x)\over n^2}.$$
This function is continuous on the line.  Much ugliness ensues when you try to show it's Hölder continuous.
