It is a common definition to say that a smooth simple closed curve $\gamma:[0,1]\rightarrow\mathbb{C}$ is "positively oriented" if the normal vector at every point along $\gamma$ points into the Jordan domain $\Gamma$ bound by $\gamma.$ I.e. for every $t\in[0,1]$ there is a $\delta>0$ such that
\begin{equation}(1) \hspace{3cm}\gamma(t)+i\epsilon\gamma'(t) \in \Gamma \quad \textrm{for every } 0<\epsilon<\delta.\end{equation}
And negatively oriented if the normals always points out.
It is easy to prove for smooth curves that if a normal vector points in (resp out), then all normal vectors point in (resp out). Thus verifying that every smooth curve is defined to be positively oriented XOR negatively oriented. And it is easy to prove that this definition agrees with a more general definition of orientation for possibly non-differentiable Jordan curves via winding number.
It is often asserted that (1) obviously extends to a definition for orientation of a piecewise smooth Jordan curve, where (1) is required to be satisfied only at all the differentiable points. For example in the setup of greens theorem most calculus texts assume this obviously constitutes a well defined notion of orientation for piecewise smooth Jordan curves. But I have yet to find a proof anywhere of this fact that if a single normal to such a curve points in, then all normals must also point in.
If the non-differentiable points are corners, then I can prove it with a straightforward argument. But I would prefer to be able to reference this fact with an existing source, rather than cluttering my paper with a lengthy proof of what most people assume to be obvious.
Question 1: does anyone know of a source for the fact that (1) well defines the orientation of a piecewise smooth Jordan curve? (It's okay if the non-differentiable points must be assumed to be corners)
Question 2: does anyone know a slick proof (<1 page) that does not require doing plane geometry on the corners?
Also, I believe a more general fact is certainly true: that is, if $\gamma:[0,1]\rightarrow\mathbb{C}$ is any Jordan curve (not necessarily smooth or piecewise smooth), that is positively oriented in the sense of winding number, then for every $t$ such that $\gamma$ is differentiable at $t$ we have that (1) holds.
When I prove this more general fact the argument gets quite elaborate, does anyone have a clever strategy, or better, a reference?