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Continuous functions

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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)

posted Apr 17, 2019 by anonymous

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