Nowhere continuous function

Short description: 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 [math]\displaystyle{ f }[/math] is a function from real numbers to real numbers, then [math]\displaystyle{ f }[/math] is nowhere continuous if for each point [math]\displaystyle{ x }[/math] there is some [math]\displaystyle{ \varepsilon \gt 0 }[/math] such that for every [math]\displaystyle{ \delta \gt 0, }[/math] we can find a point [math]\displaystyle{ y }[/math] such that [math]\displaystyle{ |x - y| \lt \delta }[/math] and [math]\displaystyle{ |f(x) - f(y)| \geq \varepsilon }[/math]. 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. This function is denoted as [math]\displaystyle{ \mathbf{1}_\Q }[/math] and has domain and codomain both equal to the real numbers. By definition, [math]\displaystyle{ \mathbf{1}_\Q(x) }[/math] is equal to [math]\displaystyle{ 1 }[/math] if [math]\displaystyle{ x }[/math] is a rational number and it is [math]\displaystyle{ 0 }[/math] if [math]\displaystyle{ x }[/math] otherwise.

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

Non-trivial additive functions

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

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

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

The Conway base 13 function is discontinuous at every point.

Hyperreal characterisation

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