The title says it all. Let $A$ be a path connected $F_\sigma$ subset of a plane (or more generally $\mathbb{R}^n$). Recall that a subset is called $F_\sigma$ if it is a union of a sequence of closed sets.

Is it true that there is a continuous surjection from $[0,1)$ onto $A$? Equivalently, can $A$ be represented as a union of an increasing sequence of Peano continuums?

Note that we cannot drop "path", since $\{y=\sin(\frac{1}{x}),x>0\}\cup \{(0,0)\}$ is a connected $F_\sigma$ subset that cannot be represented as a union of an increasing sequence of continuums.