Function from $\mathbb {R}$ to $\mathbb{R}^2$ - real analysis problem

Question

Function $f(x)$ maps $(-a,a)$ to $\mathbb{R}^2$ and $f \in C^1$ (continously differentiable). Is it possible that image of every open interval $(-b,b)$ (for $b<a$ of course) contains neighborhood of $f(0)$?

I've tried to figure out this problem, but I have no idea how to do this. I know that inverse function theorem or implicit function theorem can be helpful, anyway I don't see any way in which I could apply these theorems in this problem.

Answer 1

The answer is no. One way to show it is showing that a compact rectifiable curve in $\Bbb R ^2$ cannot contain an open disc.

Now note that if $f|_{[-b,b]}$ is continuously differentiable then necessarily it image is a rectifiable compact curve in $\Bbb R ^2$.