Nowhere continuous function

A Nowhere Continuous function is (tautologically) a function that is not Continuous at any point. That is to say, f(x) is nowhere continuous 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 , where the | | refers to absolute value. Basically, this is a statement that at each point we can choose a distance such that points arbitrarily close to our original point are taken at least that distance away.

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. This satisfies the above definition with ε =1/2 for each x because both the rational and irrational numbers are dense in the real numbers. This example is due to Dirichlet