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Wikipedia / Nowhere continuous function

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 is a function from real numbers. Therefore, no matter how close it gets to any fixed point, there are even closer points at which the function takes not-nearby values.

More general definitions of this kind of function can be obtained, by replacing the absolute value by the distance function in a metric space, or by using the definition of continuity in a topological space.

Examples

Dirichlet function

One example of such a function is the indicator function of the rational numbers, also known as the Dirichlet function and has domain and codomain both equal to the real numbers is a rational number otherwise.

is any subset of a topological space will be nowhere continuous. Functions of this type were originally investigated by Peter Gustav Lejeune Dirichlet.

Non-trivial additive functions

is called an if it satisfies Cauchy's functional equation is some constant, is additive (in fact, it is linear).

Although every linear map is also contained in some dense subset).

Discontinuous linear maps

A linear map between two topological vector spaces, such as normed spaces 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.

Other functions

Conway's base 13 function is discontinuous at every point.

Hyperreal characterisation

is nowhere continuous if its natural hyperreal is appreciable (that is, not infinitesimal).

See also

  • Blumberg theorem 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