TITLE: Hölder continuous functions are of 1st category in $C[0,1]$
QUESTION [10 upvotes]: I'm trying to show that the Hölder continuous functions in $[0,1]$ are a set of first category in $C[0,1]$. Does it suffice to show that they are not an open subset of $C[0,1]$? 
Let $\varepsilon>0$ and $f\in C[0,1]$ be Hölder. Since each Hölder continuous function is uniformly continuous, does it suffice to show that there is a non-uniformly continuous function in the ball $B_{\varepsilon}(f)\subset C[0,1]$? Can such a function be constructed or can I only prove its existence?

REPLY [2 votes]: There is a version of the open mapping theorem (e.g., in Rudin's Functional Analysis) saying that if the range a continuous linear map between Banach (or even Frechet) spaces is of second category then the map is onto (and open). It would thus be enough to find a complete norm on the space of H\"older continuous functions such that the inclusion into $C[0,1]$ is continuous.