In Greene and Krantz's Function Theory of One Complex Variable, the proof of Rouche's theorem involves the following continuity argument.
Let $f,g\colon U \to \mathbb{C}$ be holomorphic from an open set $U$. Let $\bar{D}(p,r) \subseteq U$, and \begin{equation} \vert f(\zeta) - g(\zeta) \vert \le \vert f(\zeta) + g(\zeta) \vert \end{equation} for $\zeta \in \partial D(p,r).$
Define $ f_t (\zeta) = tf(\zeta) + (1-t)\,g(\zeta), $ for $t\in [0,1]$. Then the integral $ I_t = \frac{1}{2\pi i} \oint_{\partial D(p,r)} \frac{f'_t (\zeta)}{f_t (\zeta)}\,d\zeta $ is a continuous function of $t\in [0,1]$.
They add in parenthases that the denominator does not vanish and the integrand depends continuously on $t$.
Question: Would anyone be kind enough to supply the details behind this continuity of $I_t$ ?