Dirichlet function edit

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 ${\displaystyle \mathbf {1} _{\mathbb {Q} }}$  and has domain and codomain both equal to the real numbers. By definition, ${\displaystyle \mathbf {1} _{\mathbb {Q} }(x)}$  is equal to ${\displaystyle 1}$  if ${\displaystyle x}$  is a rational number and it is ${\displaystyle 0}$  if ${\displaystyle x}$  otherwise.

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

Non-trivial additive functions edit

A function ${\displaystyle f:\mathbb {R} \to \mathbb {R} }$  is called an additive function if it satisfies Cauchy's functional equation:

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

Discontinuous linear maps edit

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 edit

The Conway base 13 function is discontinuous at every point.

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

• Blumberg theorem – even if a real function ${\displaystyle f:\mathbb {R} \to \mathbb {R} }$  is nowhere continuous, there is a dense subset ${\displaystyle D}$  of ${\displaystyle \mathbb {R} }$  such that the restriction of ${\displaystyle f}$  to ${\displaystyle D}$  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 function – a function continuous everywhere (inside its domain) and differentiable nowhere.

• "Dirichlet-function", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Dirichlet Function — from MathWorld
• The Modified Dirichlet Function Archived 2019-05-02 at the Wayback Machine by George Beck, The Wolfram Demonstrations Project.