Continuous functions

Question

X, Y are metric spaces and X is compact, f is a function mapping X into Y . Let ϕ : X ->
X x f(X) which is a subset of X x Y be defined by ϕ(x) = (x; f(x)): ϕ(X) is said to be the graph of f. Show that
f is continuous if and only if ϕ(X) is compact in the metric space X x Y with metric given by
d((x; y); (p; q)) = dX(x; p) + dY (y; q)