Define g(x) := {1/p if x=p/q for p,q relatively prime naturals, 1, x=0 or x=1, 0, otherwise} where g: [0, 1] -> [0, 1]. WTS (a) g is discontinuous at all rationals in [0,1]. (b) g is continuous at all irrationals in [0, 1].(c) Prove that g is Riemann Integrable.
(c) is pretty straightforward given a solid proof of (a)/(b) because this implies there is a countable number of discontinuities and thus has Leb. measure is 0.
However, I am a little stuck proving (a) and (b). Should I be using the density of the rationals in the reals? I am either blanking or lost and would love some help.
PS. Does anyone know how to format a piecewise properly so I can make it look better?
