I came across the following exercise:
Let $(B_t)_{t\geq 0}$ be a Brownian motion. Show that, almost surely, there is no interval $(r,s)$ on which $t\to B_t$ is Hölder continuous of exponent $\alpha$ for any $\alpha>\frac{1}{2}$. Explain the relation of this result to the differentiability properties of $B$.
I'm happy about the first part, but am wondering how this relates to the differentiability of $B$. This property alone isn't strong enough to ensure that the paths are nowhere differentiable (which they are). Does it perhaps imply that $B$ is almost surely not differentiable on any open interval?
So it would be informative to answer the question:
If $f:\mathbb{R}\to\mathbb{R}$ is not Lipschitz on any open interval, then does every open interval contain a point at which $f$ is non-differentiable?
Thank you.