Can one decompose a measurable functions as follows?

Question

Let $X$ be a metric space and $\mu$ be a Borel probability measure on $X$.

Let $f:X\rightarrow \mathbb{R}$ be a $\mu$-a.e. continuous function.

Then, does there exist a $\mu$-a.e. continuous map $g:X\rightarrow X$ and a continuous map $h:X\rightarrow \mathbb{R}$ such that $f=h\circ g$?