If $f$ is continuous and $A$ is path-connected, then $f^{-1}(A)$ path-connected

Question

If $f:\mathbb{R^n}\rightarrow\mathbb{R^m}$ is continuous and $A\subset\mathbb{R^m}$ is path-connected, then $f^{-1}(A)$ path-connected?

Answer 1

Try $n=m=1$, $f: x \mapsto x^2$ and $A = (1,\infty)$.

Answer 2

No. Let $n = m = 1$, $f \colon \mathbf R \to \mathbf R$ given by $f(x) = x^2$ and $A = \{1\}$. Then $A$ is pathwise connected, but $f^{-1}(A) = \{\pm 1\}$ is not.