My textbook reports the following result:
Given $f\colon A \subseteq \mathbb{C} \to \mathbb{C}$ holomorphic in the open simply connected set $A$, and given $\gamma\colon [a,b] \to A$ a simple closed and piecewise regular curve in $A$, then $\int_{\gamma} f(z)\,\text{d}z=0$.
Then, my textbook states that every closed piecewise regular curve is given by the union of a finite number of simple closed curves and simple open (namely non closed) curves.
Using the last fact, it expands the previous result to closed and piecewise regular curves (the relevant fact now is that there is no longer the "simple" condition).
I got it all, but I was wondering if exists a nice (and fast) manner of rigorously showing that every closed piecewise regular curve is given by the union of a finite number of simple closed curves and simple open curves.
Sorry for my english and thank you so much for your time!