Show that $w(\gamma ,0)=w(\gamma _1 ,0)+w(\gamma _2 ,0)$ $\gamma _1$ and $\gamma _2$ are two closed curves in $\Bbb C\setminus\{0\}$ both with parameter interval $[a,b]$. We define the curve $\gamma(t)=\gamma_1(t)\gamma_2(t)$.
I've just shown that $\gamma$ is closed in $\Bbb C\setminus\{0\}$. Now I have to show that $w(\gamma ,0)=w(\gamma _1 ,0)+w(\gamma _2 ,0)$. I first used a theorem saying that under the given circumstances(or so I thought), $w(\gamma,0)=(1/2\pi i)\int_\gamma (dz/z)$, which easily solved the problem. It turned out, however, that for this to be true, $\gamma$ has to be piecewise $C^1$, which we don't know if it is, so I can't use that theorem. Any ideas what to do instead? 
EDIT: My book defines winding number as $w(\gamma,0)=\frac{1}{2\pi}argvar(\gamma)$, and $argvar(\gamma)=\theta(b)-\theta(a)$, where $\theta$ is an arbitrary continuous argument function along $\gamma$.
 A: Here's a more geometric description of what's going on.
Since all the curves avoid the origin, you can parametrize them in polar coordinates using Euler's formula (treating $\mathbb{C} \setminus 0 \cong \mathbb{R}^2 \setminus \{0,0\}$).
$$
\gamma(t) = \begin{pmatrix} r(t) \cos \theta(t) \\ r(t) \sin \theta(t)\end{pmatrix},
$$
where $r$ and $\theta$ are continuous functions from $[a, b] \to \mathbb{R}$, and $r(t) > 0$ for all $t$.  Now, normalize by projecting onto the unit circle:
$$
\begin{array}{rccc}
\epsilon = \frac{\gamma}{||\gamma||}: &[a,b] &\longrightarrow &S^1 \\
& t &\longmapsto &\begin{pmatrix} \cos \theta(t) \\ \sin \theta(t)\end{pmatrix}
\end{array}
$$
This map lifts to a map $\tilde{\epsilon}: [a, b] \to \mathbb{R}$ that is unique once you specify $\tilde{\epsilon}(a)$ (or the image of any one point, for that matter).  In other words,
$$
\epsilon = p \circ \tilde{\epsilon},
$$
where $p$ is the projection $\mathbb{R} \to S^1$, $u \to (\cos u, \sin u)$.
If $\gamma$ is a closed curve, then $\epsilon$ is too, so
$$
\tilde{\epsilon}(b) - \tilde{\epsilon}(a) = 2\pi w
$$
for some $w \in \mathbb{Z}$.  This $w$ is the winding number of $\gamma$ around $0$.

Now, to address your actual question, in which you multiply the curves in the complex plane, first parametrize them as maps $[a, b] \to \mathbb{R}^2 \setminus \{0, 0\}$:
$$
\gamma_1(t) = \begin{pmatrix} r_1(t) \cos \theta_1(t) \\ r_1(t) \sin \theta_1(t)\end{pmatrix}
$$
and
$$
\gamma_2(t) = \begin{pmatrix} r_2(t) \cos \theta_2(t) \\ r_2(t) \sin \theta_2(t)\end{pmatrix}.
$$
Now,
$$
\begin{align}
\gamma(t) &= \gamma_1(t) \gamma_2(t) \\
&= r_1(t) r_2(t) \begin{pmatrix}
\cos \theta_1(t) \cos \theta_2(t) - \sin \theta_1(t) \sin \theta_2(t) \\
\sin \theta_1(t) \cos \theta_2(t) - \cos \theta_1(t) \sin \theta_2(t) 
\end{pmatrix} \\
&= r_1(t) r_2(t) \begin{pmatrix}
\cos \big(\theta_1(t) + \theta_2(t) \big) \\
\sin \big(\theta_1(t) + \theta_2(t) \big)
\end{pmatrix}.
\end{align}
$$
From this you can see that the winding number is additive.
