Nowhere continuous function explained

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.

Examples

Dirichlet function

See main article: article and 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

1 _{\Q} and has domain and codomain both equal to the real numbers. By definition, 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.^[1]

Non-trivial additive functions

See also: Cauchy's functional equation.

A function

f : \Reals \to \Reals is called an if it satisfies Cauchy's functional equation:$$f(x\; +\; y)\; =\; f(x)\; +\; f(y)\; \backslash quad\; \backslash text\; x,\; y\; \backslash in\; \backslash Reals.$$ 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 \ vert _{r} : 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 ≠ 0 , and take D : = \Q if x = 0 ).

Discontinuous linear maps

See also: Discontinuous linear functional and Continuous linear map.

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).

See also

• Blumberg theoremeven if a real function

f : \Reals \to \Reals is nowhere continuous, there is a dense subset D of \Reals such that the restriction of f to 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 functiona function continuous everywhere (inside its domain) and differentiable nowhere.

External links

• • Dirichlet Function - from MathWorld
• The Modified Dirichlet Function by George Beck, The Wolfram Demonstrations Project.

Notes and References

1. Peter Gustav . Lejeune Dirichlet . 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 . 1829 . 157–169.