Proof That a Function Continuous on Subsets of a Set is Continuous on the Union of the Sets

Theorem

Ifis a function and the restriction ofto topological subspacesand ofare both continuous and A and B are either both open or both closed, then f is continuous.

Proof

Ifandare not both closed or both open thendoes not have to be continuous. For example, let

and

and letandbe the restrictions oftoandrespectively.is not continuous becauseand this is not open.

Suppose thatandare both closed. Letbe a closed subset of

is closed inandis closed insince bothandare continuous.

so

The last expression is a union of closed subsets ofsois a closed subset ofand is continuous.