On piecewise continuous mappings of metrizable spaces

Let $f\text{\hspace{0.17em}⁣}:X\to Y$ be a resolvable-measurable mapping of a metrizable space $X$ to a regular space $Y$. Then $f$ is piecewise continuous. Additionally, for a metrizable completely Baire space $X$, it is proved that $f$ is resolvable-measurable if and only if it is piecewise continuous.