Would someone please help me with these two questions, many thx.

(a) Suppose that f : [0, 1] x [0, 1] -> R is continuous. Given x E [0, 1], prove that max{f(x, y) : y E [0, 1]} exists. (3 subpts) [i.e., prove that, with a fixed, f(x, y) achieves a maximum on [0, 1] as a function of the single variable y.] (b) Suppose that f : [0, 1] x [0, 1] -> R is continuous. By part (a), we can define g : [0, 1] -> R by g(x) = max{f(x, y) : y E [0, 1]}. Prove that g is continuous. (12 subpts)