Let $\gamma\colon [0,1]\to \mathbb{R}^2$ be a continuous **injective** map.

Is it true that for any inner point $t\in (0,1)$ there exist an open neighborhood $U$ of $\gamma(t)$ and a homeomorphism $f\colon U\tilde\to \mathbb{R}^2$ such that $f(U\cap Im(\gamma))$ is equal to a line $\mathbb{R}\subset \mathbb{R}^2$?

A reference would be helpful.