Nowhere continuous function
Template:Short description Template:More citations needed 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 <math>f</math> is a function from real numbers to real numbers, then <math>f</math> is nowhere continuous if for each point <math>x</math> there is some <math>\varepsilon > 0</math> such that for every <math>\delta > 0,</math> we can find a point <math>y</math> such that <math>|x - y| < \delta</math> and <math>|f(x) - f(y)| \geq \varepsilon</math>. 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.
ExamplesEdit
Dirichlet functionEdit
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 <math>\mathbf{1}_\Q</math> and has domain and codomain both equal to the real numbers. By definition, <math>\mathbf{1}_\Q(x)</math> is equal to <math>1</math> if <math>x</math> is a rational number and it is <math>0</math> otherwise.
More generally, if <math>E</math> is any subset of a topological space <math>X</math> such that both <math>E</math> and the complement of <math>E</math> are dense in <math>X,</math> then the real-valued function which takes the value <math>1</math> on <math>E</math> and <math>0</math> on the complement of <math>E</math> will be nowhere continuous. Functions of this type were originally investigated by Peter Gustav Lejeune Dirichlet.<ref>Template:Cite journal</ref>
Non-trivial additive functionsEdit
A function <math>f : \Reals \to \Reals</math> is called an Template:Em if it satisfies Cauchy's functional equation: <math display=block>f(x + y) = f(x) + f(y) \quad \text{ for all } x, y \in \Reals.</math> For example, every map of form <math>x \mapsto c x,</math> where <math>c \in \Reals</math> is some constant, is additive (in fact, it is linear and continuous). Furthermore, every linear map <math>L : \Reals \to \Reals</math> is of this form (by taking <math>c := L(1)</math>).
Although every linear map is additive, not all additive maps are linear. An additive map <math>f : \Reals \to \Reals</math> 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 <math>\Reals \to \Reals</math> is discontinuous at every point of its domain. Nevertheless, the restriction of any additive function <math>f : \Reals \to \Reals</math> to any real scalar multiple of the rational numbers <math>\Q</math> is continuous; explicitly, this means that for every real <math>r \in \Reals,</math> the restriction <math>f\big\vert_{r \Q} : r \, \Q \to \Reals</math> to the set <math>r \, \Q := \{r q : q \in \Q\}</math> is a continuous function. Thus if <math>f : \Reals \to \Reals</math> is a non-linear additive function then for every point <math>x \in \Reals,</math> <math>f</math> is discontinuous at <math>x</math> but <math>x</math> is also contained in some dense subset <math>D \subseteq \Reals</math> on which <math>f</math>'s restriction <math>f\vert_D : D \to \Reals</math> is continuous (specifically, take <math>D := x \, \Q</math> if <math>x \neq 0,</math> and take <math>D := \Q</math> if <math>x = 0</math>).
Discontinuous linear mapsEdit
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 functionsEdit
The Conway base 13 function is discontinuous at every point.
Hyperreal characterisationEdit
A real function <math>f</math> is nowhere continuous if its natural hyperreal extension has the property that every <math>x</math> is infinitely close to a <math>y</math> such that the difference <math>f(x) - f(y)</math> is appreciable (that is, not infinitesimal).
See alsoEdit
- Blumberg theoremTemplate:Sndeven if a real function <math>f : \Reals \to \Reals</math> is nowhere continuous, there is a dense subset <math>D</math> of <math>\Reals</math> such that the restriction of <math>f</math> to <math>D</math> is continuous.
- Thomae's function (also known as the popcorn function)Template:Snda function that is continuous at all irrational numbers and discontinuous at all rational numbers.
- Weierstrass functionTemplate:Snda function continuous everywhere (inside its domain) and differentiable nowhere.