I want to show the following.
Let $\gamma :[a,b] \to \mathbb{R}^n$ be a piecewise continuously differentiable path. Then there exists a reparametrization of $\gamma$ which is continuously differentiable.
There was a hint provided which says that you need to "slow down" at the corners to make the path differentiable at the corner. This makes sense but I don't know how I can use this. I only have the beginning like so:
" Let $\xi = \{t_{0} = a < t_{1} < \ldots < t_{K} = b\}$ be a partition of $[a,b]$ such that $\gamma _{\mid [t_{k-1},t_{k}]}$ is continuously differentiable for all $k = 1,\ldots,K$. I need to find $\psi:[\tilde a,\tilde b] \to [a,b]$ such that the reparametrization $\tilde \gamma: [\tilde a,\tilde b] \to \Bbb R^n$ with $\tilde \gamma(t)=(\gamma \circ \psi)(t)$ is continuously differentiable. "
Can somebody give me a hint how to prove this? Thanks.