Max operator preserving continuous functions

Question

Suppose you have a function $f(x,y)$ where $x \in \mathcal X$ and $y \in \mathcal Y$ such that we know for each fixed $y$, $f$ is continuous in $x$. Now define the function $g(x) = \max_{y \in Y} f(x,y)$. What would you need in order to show that $g$ is continuous in $x$?