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. Other functions The Conway base 13 function is discontinuous at every point. Hyperreal characterisation A real function is nowhere continuous if its natural hyperreal extension has the property that every is infinitely close to a such that the difference is appreciable (that is, not infinitesimal). See also Blumberg theoremeven if a real function is nowhere continuous, there is a dense subset of such that the restriction of to is continuous. Thomae's function (also known as the popcorn function)a function that is continuous at all irrational numbers and discontinuous at all rational numbers. Weierstrass functiona function continuous everywhere (inside its domain) and differentiable nowhere. References External links Dirichlet Function — from MathWorld The Modified Dirichlet Function by George Beck, The Wolfram Demonstrations Project. Mathematical analysis Topology Types of functions