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

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

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

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

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

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.

Conway's base 13 function is discontinuous at every point.

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

• Blumberg theorem – even if a real function ${\displaystyle f:\mathbb {R} \to \mathbb {R} }${\displaystyle f:\mathbb {R} \to \mathbb {R} } is nowhere continuous, there is a dense subset ${\displaystyle D}${\displaystyle D} of ${\displaystyle \mathbb {R} }${\displaystyle \mathbb {R} } such that the restriction of ${\displaystyle f}${\displaystyle f} to ${\displaystyle D}${\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年05月02日 at the Wayback Machine by George Beck, The Wolfram Demonstrations Project.

• Mathematical analysis
• Topology
• Types of functions

• Articles with short description
• Short description is different from Wikidata
• Articles needing additional references from September 2012
• All articles needing additional references
• Webarchive template wayback links