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.

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 $$0$$ if $$x$$ 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: $$f(x + y) = f(x) + f(y) \quad \text{ for all } x, y \in \Reals.$$ 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
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).