Are the connected components of the level sets of a $\mathcal{C}^1$ function path-connected?

Question

I have a $\mathcal{C}^1$ (or even just $\mathcal{C}^0$) function $f:\mathbb{R}^n\rightarrow\mathbb{R}$, and have been trying to figure out when the connected components of its levels sets are also path-connected.

With such a well-behaved function, I had initially suspected they would be path-connected, but haven't been able to attack it.
The only tool I really have is the fact that the continuous image of a path-connected set is path-connected, which doesn't seem to get me very far.

Any help is greatly appreciated.

Answer 1

Consicer $f(x,y)=\begin{cases}x^2(y-\sin\frac1x)&x\ne0\\0&x=0\end{cases}$. Then the level set $f(x,y)=0$ is connected but not pathconnected.