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 f is a function from real numbers to real numbers, then f is nowhere continuous if for each point x there is an ε > 0 such that for each δ > 0 we can find a point y such that 0 < |x − y| < δ and |f(x) − f(y)| ≥ ε. Therefore, no matter how close we get 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, named after German mathematician Peter Gustav Lejeune Dirichlet. This function is denoted as IQ and has domain and codomain both equal to the real numbers. IQ(x) equals 1 if x is a rational number and 0 if x is not rational. If we look at this function in the vicinity of some number y, there are two cases:
- If y is rational, then f(y) = 1. To show the function is not continuous at y, we need to find an ε such that no matter how small we choose δ, there will be points z within δ of y such that f(z) is not within ε of f(y) = 1. In fact, 1/2 is such an ε. Because the irrational numbers are dense in the reals, no matter what δ we choose we can always find an irrational z within δ of y, and f(z) = 0 is at least 1/2 away from 1.
- If y is irrational, then f(y) = 0. Again, we can take ε = 1/2, and this time, because the rational numbers are dense in the reals, we can pick z to be a rational number as close to y as is required. Again, f(z) = 1 is more than 1/2 away from f(y) = 0.
In less rigorous terms, between any two irrationals, there is a rational, and vice versa.
The Dirichlet function can be constructed as the double pointwise limit of a sequence of continuous functions, as follows:
for integer j and k.
In general, 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.
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 (i.e., not infinitesimal).
- P.G., Lejeune Dirichlet (1829). "Sur la convergence des séries trigonométriques qui servent à répresenter une fonction arbitraire entre des limites donées. [On the convergence of trigonometric series which serve to represent an arbitrary function between given limits]". Journal für die reine und angewandte Mathematik. 4: 157–169.
- Dunham, William (2005). The Calculus Gallery. Princeton University Press. p. 197. ISBN 0-691-09565-5.
- Hazewinkel, Michiel, ed. (2001) , "Dirichlet-function", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
- Dirichlet Function — from MathWorld
- The Modified Dirichlet Function by George Beck, The Wolfram Demonstrations Project.