= Solution
Let $u(t)=\gamma(t)/|\gamma(t)|$, a <continuous map> into the <unit circle>. By <uniform continuity>, partition $[0,1]$ so finely that on each interval beginning at $t_j$, $|u(t)/u(t_j)-1|<1$. This ratio lies in the right half-plane and has a continuous <complex argument> with value zero at $t_j$. Starting from $\theta_0$, add these local arguments successively, matching endpoint values. This constructs a continuous real lift with $u(t)=e^{i\theta(t)}$.
Any two such lifts with the same initial value differ by a <continuous function> taking values in $2\pi\mathbb Z$, and hence coincide. Since the curve closes, $e^{i(\theta(1)-\theta(0))}=1$. Define its <winding number> by
$$
\boxed{w(\gamma)=\frac{\theta(1)-\theta(0)}{2\pi}\in\mathbb Z.}
$$
Changing the chosen initial argument adds a constant multiple of $2\pi$ to the whole lift, so the <winding number> is independent of that choice. No <differentiability> of the curve is required.
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