Here is a function with strange continuity properties: (a) Show that is discontinuous at c if c is rational. There exist irrational numbers arbitrarily

Here is a function with strange continuity properties:

(a) Show that ƒ is discontinuous at c if c is rational. There exist irrational numbers arbitrarily close to c.
(b) Show that ƒ is continuous at c if c is irrational. Let I be the interval {x : |x − c| 0, I contains at most finitely many fractions p/q with q

1 if x is the rational number p/q in f(x)= q lowest terms 0 if x is an irrational number