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 is a function from real numbers to real numbers, then is nowhere continuous if for each point there is some such that for every we can find a point such that and . 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   and has domain and codomain both equal to the real numbers. By definition,   is equal to   if   is a rational number and it is   if   otherwise.

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

Non-trivial additive functions


A function   is called an additive function if it satisfies Cauchy's functional equation:   For example, every map of form   where   is some constant, is additive (in fact, it is linear and continuous). Furthermore, every linear map   is of this form (by taking  ).

Although every linear map is additive, not all additive maps are linear. An additive map   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   is discontinuous at every point of its domain. Nevertheless, the restriction of any additive function   to any real scalar multiple of the rational numbers   is continuous; explicitly, this means that for every real   the restriction   to the set   is a continuous function. Thus if   is a non-linear additive function then for every point     is discontinuous at   but   is also contained in some dense subset   on which  's restriction   is continuous (specifically, take   if   and take   if  ).

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   is nowhere continuous if its natural hyperreal extension has the property that every   is infinitely close to a   such that the difference   is appreciable (that is, not infinitesimal).

See also

  • Blumberg theorem – even if a real function   is nowhere continuous, there is a dense subset   of   such that the restriction of   to   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.


  1. ^ Lejeune Dirichlet, Peter Gustav (1829). "Sur la convergence des séries trigonométriques qui servent à représenter une fonction arbitraire entre des limites données". Journal für die reine und angewandte Mathematik. 4: 157–169.