How big is the set of Hölder continuous functions?
Solution 1:
The space of polynomials is dense in $C^0$. Since polynomials are Lipschitz on $[0,1]$, they are a fortiori $\delta$-Hölder continuous for any $\delta \in (0,1)$. We conclude that $\bigcup_{\delta \in (0,1)}C^\delta$ (or in fact, any individual $C^\delta$) is dense in $C^0$.