Given a piecewise $C^1$ Jordan curve $\sigma$, there is a region $A$ bounded by $\sigma$. Is it true that $\overline A$ is $C^1$-diffeomorphic to a closed polygon? (I believe this is the same as claiming $\overline A$ has a $C^1$ triangulation.)
A curve $\phi: [0,1]\to \mathbb R^2$ is said to be piecewise $C^1$ if there exist finitely many $a_i$ that $0 = a_0 < a_1 < \cdots < a_{n-1} < a_n = 1$, and $\phi$ is $C^1$ on each $[a_i, a_{i+1}]$.
The claim holds if $\sigma$ is $C^1$ (not just piecewise $C^1$) as is shown in this question.
I am looking for a rigorous proof of Green's theorem which is said to hold for piecewise $C^1$ Jordan curves.