Let f be a bounded function on [a, b], which is continuous everywhere except finitely many points X1, X2, follows. We'll use induction on N. XN. We'll prove that f is integrable as (a) Base case: N = 1. So f has only one discontinuity in [a, b] - call it x. We'll simplify our proof by assuming that x = (a, b). Let > 0. We'll show that there exists a partition Q of [a, b] so that UQ(f) - Lo(f) < (i) Let 0 < a < min{a-xil. lb-x11). Choose a partition P = P U P where P is a partition of [a, x-a] and P2 is a partition of [x+a, b]. Sketch such a partition. (ii) Let M = sup{f(x) | xe [x - a x+a]) and m = inf{f(x) | xe [x-a, x+a]). Show that you can choose a so that MAX - MAX SE/3, where Ax is the length of the interval [x-a, x + a]. (iii) Use the fact that f is continuous on [a, x-a] and on [x+a, b] to show that we can choose P1, P2 so that Up (f) - Lp1 (f)