Reparameterization of Directed Smooth Curve Preserves Image

Theorem

Let $\closedint a b$ and $\closedint c d$ be closed real intervals.

Let $\gamma : \closedint a b \to \C$ be a smooth path.

Suppose that $\sigma : \closedint c d \to \C$ is a reparameterization of $C$.

Then $\Img \gamma = \Img \sigma$, where $\Img \gamma$ denotes the image of $\gamma$.

By definition of directed smooth curve, there exists a bijective differentiable strictly increasing function $\phi: \closedint c d \to \closedint a b$ such that $\sigma = \gamma \circ \phi$.

From Surjection by Restriction of Codomain, it follows that there exists a function $\tilde \gamma: \closedint a b \to \Img \gamma$ that is the surjective restriction of $\gamma$ to $\Img \gamma$.

Put $\tilde \sigma := \tilde \gamma \circ \phi : \closedint c d \to \Img \gamma$.

Then:

$\Img {\tilde \sigma} = \Img \gamma$

Now:

$\map \sigma t = \map {\tilde \sigma} t$ for all $t \in \closedint c d = \Dom \sigma = \Dom {\tilde \sigma}$

where $\Dom \sigma$ denotes the domain of $\sigma$.

It follows that:

$\Img \sigma = \Img {\tilde \sigma}$

Hence:

$\Img {\tilde \gamma} = \Img \sigma$

$\blacksquare$