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.
is a function from real numbers to real numbers, then
we can find a point
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.
and has domain and codomain both equal to the real numbers.
is a rational number and it is
is any subset of a topological space
then the real-valued function which takes the value
Functions of this type were originally investigated by Peter Gustav Lejeune Dirichlet.
is called an additive function if it satisfies Cauchy's functional equation:
For example, every map of form
is some constant, is additive (in fact, it is linear and continuous).
Furthermore, every linear map
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
is a non-linear additive function then for every point
is also contained in some dense subset
is continuous (specifically, take
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.
Every linear functional is a linear map and on every infinite-dimensional normed space, there exists some discontinuous linear functional.
The Conway base 13 function is discontinuous at every point.
is nowhere continuous if its natural hyperreal extension has the property that every
is appreciable (that is, not infinitesimal).