Is connected component of sublevel set of continuous function always path connected?

Question

Let $f:\mathbb{R}^n\mapsto\mathbb{R}$ be a Lipschitz continuous function. Let $S_a\triangleq \{x\in\mathbb{R}^n\,|\,f(x)\leq a\}$ be a sublevel set of $f$. If $S_a$ is connected, is it always path-connected? If not, any counterexample?

Answer 1

Let $A\subseteq\Bbb R^2$ be closed and connected but not path connected (for example you can take as $A$ the $\sin(1/x)$-continuum, also called the topologist's sine curve).

Consider $f(x)=d(x,A)$ and look at $S_0$.