Let $\gamma : [0,1] \rightarrow \mathcal{M}$ be a continuous map so that $[0,1]$ is homeomorphic to $\gamma([0,1])$, where $\mathcal{M}$ is a manifold (Hausdorff, second countable, and locally Euclidean). Using a chart containing $\gamma(0)$, I think it is always possible to find a circle centered at $\gamma(0)$ that intersects the curve $\gamma([0,1])$ at a single point. Can someone help me prove this?

The intuition I have is that it should be possible to choose a radius $r$ small enough so that $\gamma([0,1])$ intersects the circle exactly once. However I'm not sure how to reason that the curve is not like a "space filling curve" locally around $\gamma(0)$. I know this has to do something with $[0,1]$ (hence $\gamma([0,1])$) being compact, but not sure how the argument should go.