# Almost sure existence of 1/2-Hölder time points of Brownian motion

Supposedly (I cannot find any reference, hence this question), it should hold for the standard Brownian motion $B=(B_t)_{t\ge0}$ that for almost every $\omega \in \Omega$ there exists a $t\ge 0$ such that $B(\omega)$ is 1/2-Hölder continuous at $t$.

I was trying to look that up in various sources, including ̈Mörters' and Peres' book on Brownian Motion but to no luck. Could you direct me to any source for that claim (or alternatively one that disproves it)?

Since we have almost surely $\alpha$-Hölder continuity everywhere for $\alpha<1/2$ and nowhere for $\alpha>1/2$ (as well as nowhere in any countable subset of $\mathbb{R}_+$ for $\alpha=1/2$), I assume the argument for showing the claim has to be quite fine.