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 entire continuity definition by the definition of continuity in a topological space.

On example of such a function is a function *f* on the real numbers such that *f(x)* is 1 if *x* is a rational number, but 0 if *x* is not rational. If we look at this function in the vincinity 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 a single *ε* which works in the above definition. In fact, 1/2 is such an *ε*, since we can find an irrational number *z*arbitrarily close to y and *f(z)*=0, 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 we pick *z* to be an rational number as close to *y* as is required. Again, *f(z)* is more than 1/2 away from *f(y)*

The discontinuities in this function occur because both the rational and irrational numbers are dense in the real numbers. It was originally investigated by Dirichlet.)