Function continuous in one variable, Borel in other is jointly Borel?

Question

Suppose I have a function $f: X\times Y\to \mathbb{C}$, where $X$ and $Y$ are compact Hausdorff spaces, where the function is continuous in $X$ and Borel in $Y$. Is it true that the function is jointly Borel in $X\times Y$? What additional conditions might ensure this?

Thanks.