Reparameterization of Directed Smooth Curve Maps Endpoints To Endpoints/Complex Plane

Theorem
Let $\C$ denote the complex plane. Let $\closedint a b$ and $\closedint c d$ be closed real intervals. Let $\gamma: \closedint a b \to \C$ be a smooth path in $\C$.

Let $C$ be a directed smooth curve with parameterization $\gamma$.

Let $\sigma: \closedint c d \to \C$ be a reparameterization of $C$.

Then the start points and end points of $\gamma$ and $\sigma$ are identical:


 * $\map \gamma a = \map \sigma c$


 * $\map \gamma b = \map \sigma d$

Proof
By definition of reparameterization, there exists a bijective differentiable strictly increasing real function $\phi: \closedint c d \closedint a b$ such that $\sigma = \gamma \circ \phi$.

As $\map {\phi^{-1} }{a} \in \closedint c d$:
 * $c \le \map {\phi^{-1} }{a}$

As $\phi$ is strictly increasing:
 * $\map \phi c \le \map \phi { \map {\phi^{-1} } a } = a$

As $\map \phi c \in \closedint a b$:
 * $\map \phi c = a$

Hence:


 * $\map \sigma c = \map {\gamma \circ \phi} c = \map \gamma a$

As $\map {\phi^{-1} } b \in \closedint c d$:
 * $d \ge \map {\phi^{-1} } b$

As $\phi$ is strictly increasing:
 * $\map \phi d \ge \map \phi { \map {\phi^{-1} } b } = b$

As $\map \phi d \in \closedint a b$:
 * $\map \phi d = b$

Hence:


 * $\map \sigma d = \map {\gamma \circ \phi} d = \map \gamma b)$