We know by Rademacher's theorem that a Lipschitz-continuous map is differentiable almost everywhere (a.e.).
Now for every path in $\mathbb{R}^n$, $n>1$, there is a Lipschitz-continuous map such that it will be nowhere differentiable on that path. But what about a given Lipschitz map. Can we find a path between every pair of points where it is a.e. differentiable?
Precisely, let $f:\mathbb{R}^n \to \mathbb{R}$ be a Lipschitz map with $n>1$. For every $x,y\in \mathbb{R}^n$, is it always possible to find a continuously differentiable path $\gamma: [0,1] \to \mathbb{R}^n$, with $\gamma(0)=x$ and $\gamma(1)=y$, such than $f(\gamma(t))$ is differentiable a.e. on $[0,1]$?