Define by

(a) Show that is discontinuous at every rational number.

So there exists a sequence converging to , such that does not converge to . So maybe come up with a sequence that has both irrational and irrational terms? Thus it would oscillate? For example, we can approximate by a sequence with both rational and irrational terms?

(b) Fix and . Show that maps only finitely many elements of to .

Suppose and . In other words, we want to show that maps only finitely many terms from to . So given a sequence that approximates , it will have a finite number of rational terms from part (a). Is this correct?

(c) Show that is continuous at every irrational number.

Given any sequence in converging to , , converges to . So we can always find an "irrational" sequence that approximates ? Thus converges to ?

(d) Let . Prove that maps only finitely many elements of to a value greater than .

This is the same as part(b) right?

(e) Prove that is Riemann integrable on and that .

Here we just consider ? And bound it and show it is less than ?