Proof That a Function Continuous on Subsets of a Set is Continuous on the Union of the Sets
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.
Ifandare not both closed or both open thendoes not have to be continuous. For example, let
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.
The last expression is a union of closed subsets ofsois a closed subset ofand is continuous.