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Conditions For a Discontinuous Function to be Discontinuous on a Subtopology

Theorem

Supposeis not a continuous function.

Ifis coarser thanso thatandis finer thanso thatthenis also not continuous.

Proof

Sinceis not continuous, there is an open setsuch thatis not an element of

Sinceandwe have

Sinceand since

Henceis not continuous.