Finding an example of "A reparametrization of a closed curve need not be closed"$\text{}$

Question

I look at the following exercise of the book "Elementary Differential Geometry" of Andrew Pressley:

"Give an example to show that a reparametrization of a closed curve need not be closed."

Any hints how we could do that? I don't have any idea how to find such an example.

Answer 1

Pressley's definition is

A non-constant smooth curve $\gamma : \mathbb R \to \mathbb R^n$ is closed if there is a real $T \ne 0$ such that $\gamma(t+T) = \gamma(t)$ for all $t$.

Consider the circle $\gamma(t) = (\cos t, \sin t)$. This is closed (with period $T= 2\pi$), but the reparametrization $\gamma(t^3+t)$ is not, since it is not periodic.