Proof That a Function is Continuous if and Only if the Inverse Image of a Closed Set is Closed
A functionis continuous if and only if the inverse image of every closed subset ofis closed in
Ifis continuous then
Supposeis continuous andis any closed subset ofthenis an open subset ofandis an open subset of
Suppose for everyis closed inandis closed in
Letbe an open subset ofthenis closed inandis closed in
Henceis an open subset ofandis continuous.