Winding number of a continuous closed path
= Winding number of a continuous closed path
If $\gamma:[0,1]\to\mathbb C\setminus\{0\}$ is <continuous function>[continuous] and closed, choose a continuous lift $\theta$ such that
$$
\frac{\gamma(t)}{|\gamma(t)|}=e^{2\pi i\theta(t)}.
$$
Then $\operatorname{wind}(\gamma,0)=\theta(1)-\theta(0)\in\mathbb Z$. This agrees with the contour-integral definition when the path is piecewise smooth.