Pathology

There are some pathological functions which are constructed by assigning one set of values to all rational numbers in an interval and another set of values to all irrational numbers in the interval. Such functions are usually constructed as examples of functions that are not integrable in the Riemann sense.

Here's an interesting example that is integrable in the usual sort of Riemann or Darboux sense. Define f : [1,2] &rarr R. For x in [1,2], let f(x) = 0 if x is irrational. Or let f(x)=1/n if x is rational, where x=m/n expressed in lowest terms.