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Nowhere continuous

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 . 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. 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 zarbitrarily 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.)