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. 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 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 plainer 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.
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 Dirichlet.
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