 Solution

For $0\le t\le 1$, put $F_t=f+tg$. On the curve, $|F_t|\ge |f|-t|g|>0$, so no member of this homotopy has a boundary zero. The argument principle counts the interior zeros, with multiplicity, by its winding number $N(t)=\mathrm{wind}(F_t\circ \gamma ,0)$. For a piecewise smooth curve this isThere are no poles, and the null-homotopy of the Jordan curve in the domain permits this interior count. The loops $F_t\circ \gamma$ form a continuous homotopy avoiding zero, so their integer winding number stays constant. Equivalently, for a piecewise smooth curve the integral varies continuously with $t$ and is integer-valued. In particularThis proves Rouché's theorem. The counting is with multiplicity and assumes the usual positive orientation; reversing orientation reverses both winding counts without changing the conclusion.