# 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*(*x*) 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.

## Dirichlet function

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.^{[1]} This function is written *I*_{Q} and has domain and codomain both equal to the real numbers. *I*_{Q}(*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*.

This shows that the Dirichlet function is a Baire class 2 function. It cannot be a Baire class 1 function because a Baire class 1 function can only be discontinuous on a meagre set.^{[2]}

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.

## 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 (i.e., not infinitesimal).

## See also

- Thomae's function (also known as the popcorn function) — a function that is continuous at all irrational numbers and discontinuous at all rational numbers.

## References

- ↑ Lejeune Dirichlet, P. G. (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 reine und angewandte Mathematik*[Journal for pure and applied mathematics (also known as*Crelle's Journal*)], vol. 4, pages 157–169. - ↑ Dunham, William (2005).
*The Calculus Gallery*. Princeton University Press. p. 197. ISBN 0-691-09565-5.

## External links

- Hazewinkel, Michiel, ed. (2001), "Dirichlet-function",
*Encyclopedia of Mathematics*, Springer, ISBN 978-1-55608-010-4 - Dirichlet Function — from MathWorld
- The Modified Dirichlet Function by George Beck, The Wolfram Demonstrations Project.