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. 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 $, x - y, < \delta$ 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

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 $\mathbf_\Q$ and has domain and codomain both equal to the real numbers. By definition, $\mathbf_\Q(x)$ 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 if it satisfies Cauchy's functional equation:
$f(x + y) = f(x) + f(y) \quad \text x, y \in \Reals.$
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 $f\big\vert_ : r \, \Q \to \Reals$ to the set $r \, \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

The Conway 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).

See also

* Blumberg theoremeven if a real function $f : \Reals \to \Reals$ is nowhere continuous, there is a dense subset $D$ of $\Reals$ such that the restriction of $f$ to $D$ 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
{{Webarchive, url=https://web.archive.org/web/20190502165330/http://demonstrations.wolfram.com/TheModifiedDirichletFunction/ , date=2019-05-02 by George Beck, The Wolfram Demonstrations Project.

Mathematical analysis
Topology
Types of functions