Nowhere continuous function

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title: "Nowhere continuous function" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["mathematical-analysis", "topology", "types-of-functions"] description: "Function which is not continuous at any point of its domain" topic_path: "science/mathematics" source: "https://en.wikipedia.org/wiki/Nowhere_continuous_function" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Function which is not continuous at any point of its domain ::

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. If f is a function from real numbers to real numbers, then f is nowhere continuous if for each point x there is some \varepsilon 0 such that for every \delta 0, we can find a point y such that |x - y| and |f(x) - f(y)| \geq \varepsilon. 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

Main article: Dirichlet function

One example of such a function is the indicator function of the rational numbers, also known as the Dirichlet function. This function is denoted as \mathbf{1}\Q and has domain and codomain both equal to the real numbers. By definition, \mathbf{1}\Q(x) is equal to 1 if x is a rational number and it is 0 otherwise.

More generally, if E is any subset of a topological space X such that both E and the complement of E are dense in X, then the real-valued function which takes the value 1 on E and 0 on the complement of E will be nowhere continuous. Functions of this type were originally investigated by Peter Gustav Lejeune Dirichlet.

Non-trivial additive functions

A function f : \Reals \to \Reals is called an additive function if it satisfies Cauchy's functional equation: f(x + y) = f(x) + f(y) \quad \text{ for all } x, y \in \Reals. For example, every map of form x \mapsto c x, where c \in \Reals is some constant, is additive (in fact, it is linear and continuous). Furthermore, every linear map L : \Reals \to \Reals is of this form (by taking c := L(1)).

Although every linear map is additive, not all additive maps are linear. An additive map f : \Reals \to \Reals is linear if and only if there exists a point at which it is continuous, in which case it is continuous everywhere. Consequently, every non-linear additive function \Reals \to \Reals is discontinuous at every point of its domain. Nevertheless, the restriction of any additive function f : \Reals \to \Reals to any real scalar multiple of the rational numbers \Q is continuous; explicitly, this means that for every real r \in \Reals, the restriction f\big\vert_{r \Q} : r , \Q \to \Reals to the set r , \Q := {r q : q \in \Q} is a continuous function. Thus if f : \Reals \to \Reals is a non-linear additive function then for every point x \in \Reals, f is discontinuous at x but x is also contained in some dense subset D \subseteq \Reals on which f's restriction f\vert_D : D \to \Reals is continuous (specifically, take D := x , \Q if x \neq 0, and take D := \Q if x = 0).

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

A real function f is nowhere continuous if its natural hyperreal extension has the property that every x is infinitely close to a y such that the difference f(x) - f(y) is appreciable (that is, not infinitesimal).

References

References

1. Lejeune Dirichlet, Peter Gustav. (1829). "Sur la convergence des séries trigonométriques qui servent à représenter une fonction arbitraire entre des limites données". Journal für die reine und angewandte Mathematik.

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