Single Point Characterization of Simple Closed Contour/Lemma 3

Lemma
Let $\tilde \gamma_1 : \closedint a b \to \C$ be an injective piecewise continuously differentiable function such that $\map {\tilde \gamma_1 '}{t} \ne 0$ for all $t \in \openint a b$.

Let $\tilde t_3, \tilde t_4 \in \openint a b$ with $\tilde t_3 < \tilde t_4$.

Let $\tilde t_5 \in \openint {\tilde t_3}{\tilde t_4}$ such that $\tilde \gamma_1$ is complex-differentiable at all $t \in \openint a b \setminus \set { \tilde t_5 }$.

Let $r_1 \in \R_{>0}$ such that:


 * $r_1 = \cmod{ \map {\tilde \gamma_1}{\tilde t_3} - \map {\tilde \gamma_1}{\tilde t_5} } = \cmod { \map {\tilde \gamma_1}{\tilde t_4} - \map {\tilde \gamma_1}{\tilde t_5} }$

Let $\delta \in \R_{>0}$ such that for all $\epsilon \in \openint 0 \delta$:


 * $\map {\tilde \gamma_1}{\tilde t_3 + \epsilon}, \map {\tilde \gamma_1}{\tilde t_4 - \epsilon} \in \map { B_{r_1} }{ \map {\tilde \gamma_1}{\tilde t_5} }$

where $\map { B_{r_1} }{ \map {\tilde \gamma_1}{\tilde t_5} }$ denotes an open disk.

Then there exists an injective complex-differentiable function $\tilde \gamma_2 : \closedint a b \to \C$ such that:


 * $\map {\tilde \gamma_2} t = \map {\tilde \gamma_1}{t}$ for all $t \in \closedint {a}{\tilde t_3} \cup \closedint {\tilde t_4}{b}$
 * $\map {\tilde \gamma_2} t \in \map {B_{r_1} }{ \map {\tilde \gamma_1}{\tilde t_5} }$ for all $t \in \openint{\tilde t_3}{\tilde t_4}$