Theorem 10.12 from Rudin's Real and Complex Analysis says:
Suppose $F \in H(\Omega),F'$ is continuous in $\Omega$ .Then for every closed path $\gamma$ in $\Omega$ ,$\int_{\gamma}F'(z)dz=0$.
And a path is defined to be a piecewise continuously differentiable curve in the plane.
The key point of the proof is to use the fundamental theorem of calculus:
$\int_\gamma F'(z)dz:=\int_{\alpha}^{\beta} F'(\gamma(t))\gamma'(t)dt\color{red}{=} F(\gamma(\beta))-F(\gamma(\alpha))=0$ .
Question:
Since $\gamma$ is piecewise continuously differentiable,then $F'(\gamma(t))\gamma'(t)$ is not continuous on some points in $[\alpha,\beta]$. $F'(\gamma(t))\gamma'(t)$ could have no primitive functions and we cannot use the fundamental theorem of calculus.
So I think maybe closed path $\gamma$ should be revised as closed path $\gamma$ which is continuously differentiable?