Let $C^\delta$ be all $\delta$-Hölder continuous functions on $[0,1]$. Is $\cup_{\delta\in (0,1)} C^\delta$ a dense subset of $C^0$, the set of continuous functions on $[0,1]$ in the uniform metric?
In other words, if $f\in C^0$, can we find a sequence of Hölder continuous functions $f_n$ such that
$$\sup_{x\in [0,1]} |f_n – f| \to 0, \hbox{ as } n\to \infty?$$
Best Answer
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 $C^0$.