In mathematics, a nowhere continuous function, also called an everywhere discontinuous function, is a function that is not continuous at any point of its domain.

Also, "monsters" (nowhere continuous functions, continuous but nowhere differentiable functions, space-filling curves) began to be investigated.

As a consequence of the Stone-Weierstrass theorem, the graph of this operator is dense in XxY, so this provides a sort of maximally discontinuous linear map (confer nowhere continuous function).