Connected components of intersection of curve with Ball

Question

Given a smooth curve $\gamma\colon\left[0,1\right]\to\mathbb R^n$, does the intersection $\mathrm{im}\,\gamma\cap B_1^n\left(0\right)$ have finitely many connected components? (Here $B^n_1\left(0\right)$ is the unit ball in $\mathbb R^n$ centred at $0\in\mathbb R^n$).

Answer 1

Not necessarily. Let $\gamma(t)=(t,-\sqrt{1-t^2}+e^{\frac{-1}t}\sin\frac1t)$, a smooth curve in the plane, with $\mathrm{im}\,\gamma$ same as the graph of $f(t)=-\sqrt{1-t^2}+e^{\frac{-1}t}\sin\frac1t$, $0\le t\le1$. The idea is that it crosses the boundary of the unit disk infinitely many times as $t\to0$.