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QUESTION

# Let (X, dX) and (Y, dY ) be metric spaces, and let f : X Y be continuous. Define the distance d on the product space X Y as in class, so that (X Y,...

Let (X, dX) and (Y, dY ) be metric spaces, and let f : X → Y be continuous. Define the distance d on the product space X × Y as in class, so that (X × Y, d) is a metric space. Show that the graph Γf of f, defined by Γf = {(x, y) ∈ X × Y | f(x) = y}, is a closed subset of X × Y .