# Nowhere continuous

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 |

*x*−

*y*|<

*δ*and |

*f*(

*x*)−

*f*(

*y*)|≥

*ε*. The import of this statement is that no matter how close we get to any fixed point, there are nearby 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 the continuity definition by the definition of continuity in a topological space.

One example of such a function is the indicator function of the rational numbers. This function is written *I*_{Q} and has domain and range 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 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.

*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 Dirichlet.