A question from E.M Stein's book. How to analyze the Continuity of $z_{\epsilon}$

Question

This is a problem from Stein's Complex Analysis, $f,g$ are holomorphic on $|z|≤1$, and $f(0)=0$,and vanishes nowhere on the unit disc. suppose $z_{\epsilon}$ is the zero of function $f_{\epsilon}=f(z)+\epsilon g(z)$,show that $\epsilon\rightarrow z_{\epsilon}$ is continous. Well,the existence of zeros comes directly from the Rouché's Theorem,but for the continuity,I guess it needs a topological proof .searching for help