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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.

## 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 *I*_{Q} or *1*_{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.

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]}

## 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

- Blumberg theorem – even if a real function
*f*: ℝ → ℝ is nowhere continuous, there is a dense subset*D*of ℝ 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 function – a function
*continuous*everywhere (inside its domain) and*differentiable*nowhere.

## References

**^**Lejeune Dirichlet, Peter Gustav (1829). "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**: 157–169.