I failed to give an appropriate title to the question, so any suggestion for a better title is welcome: Here's the question: I would like to prove the following result: Given $\varepsilon>0$, a Lipschitz map $u:\mathbb R^n \to \mathbb R^m$ and two distinct points $p,q\in \mathbb R^n$, there exists a curve $\gamma:[0,1]\to\mathbb R^n$ with $\gamma(0)=p, \gamma(1)=q$ and such that $$ \int_0^1|\gamma'(t)| \mathrm dt < |p-q|+\varepsilon $$ and such that $(u\circ \gamma)'=Du (\gamma')$ $\mathcal L^1$-almost everywhere in $[0,1]$, and $\mathcal L^1$ denotes the one-dimensional Lebesgue-measure on $[0,1]$.
EDIT: The curve $\gamma$ should be Lipschitz, but if the result holds for smooth $\gamma$ as well this is of course fine.
EDIT for a better explanation: $(u\circ \gamma)$ is a.e. differentiable by Rademachers Theorem. However, the equality $(u\circ \gamma)'=Du (\gamma')$ need a priori not to hold a.e. in $[0,1]$ since $Du$ is only defined $\mathcal L^n$-a.e. a priori. So the result says essentially that this can be fixed with a small perturbation.