If $ f: X times Y to mathbb {R} $ is continuous and $ X $ compact, is $ and mapsto sup_x f (x, y) $ continuous?

Suppose $ X, Y $ they are metrically spaced and $ X $ compact. Yes $ f: X times Y to mathbb {R} $ it is continuous, it is $ y mapsto sup_ {x in X} f (x, y) $ A continuous function?