Editing Nowhere continuous function (section)

===Discontinuous linear maps===

{{See also|Discontinuous linear functional|Continuous linear map}}

A [[linear map]] between two [[topological vector space]]s, such as [[normed space]]s for example, is continuous (everywhere) if and only if there exists a point at which it is continuous, in which case it is even [[uniformly continuous]]. Consequently, every linear map is either continuous everywhere or else continuous nowhere.
Every [[linear functional]] is a [[linear map]] and on every infinite-dimensional normed space, there exists some [[discontinuous linear functional]].