Variation of Argument :
Definition( Collect from my book ) : Let $f$ be analytic inside and on a sinple closed contour $C$ except possibly for poles inside $C$ and $f(z)\not=0$ on $C$. As $z$ describes $C$ once in the positive direction in the $z$-plane , the image point $w=f(z)$ describes a closed curve $\Gamma=f(C)$ in the $w$-plane in a particular direction which determines the orientation of the image curve $\Gamma$. Since $f(z)\not=0$ on $C$ , $\Gamma$ never passes through the origin in the $w$-plane. Let $w_0$ be the arbitrary fixed point on $\Gamma$ and let $\phi_0$ be the argument of $w_0$. Then let , $\arg z$ run continuously from $\phi_0$ , as the point begins at $w_0$ and traverses $\Gamma$ once in the direction of orientation assigned to it by $w=f(z)$. If $w$ returns to the staring point $w_0$ , then $\arg w$ assume a particular value of $\arg w_0$ which we denote by $\phi_1$. We define , $$\Delta_C\arg f(z)=\phi_1-\phi_0.$$
I am unable to understand the meaning of bold sentences. Can anyone explain these sentences with a proper example or by a rough figure such that I can realize what actually the variation of argument $\Delta_C f(z)$ over $C$?
If you can provide an example of a function and a simple closed contour $C$ explaining the value of $\Delta_C f(z)$ over $C$ then it is better to me.