1. For every part of the problem, let a,b be real numbers and a < b.

(a) Consider a function f : [a, b] → R, and assume that there is a single point at which f is nonzero,

i.e. assume that there is a point c € [a, b] so that f satisfies f(x) = 0 for all x € [a, b]\ {c}. Prove

that f is integrable on [a, b] and that fo f(x)dx = 0.

(b) Let f: [a, b] → R be an integrable function. Let g: [a, b] → R be a function which agrees with f

at all points in [a, b] except for one, i.e. assume there exists a c € [a, b] such that g(x) = f(x) for

all x € [a, b]\ {c}. Prove that g is integrable on [a, b] and that få g(x) dx = f f(x)dx. Hint: you

may need to use the result from part (a).

g(r)dr

(c) Let f: [a, b] → R be an integrable function. Let g: [a, b] → R be a function which agrees with f at

all points in [a, b] except on a finite set, i.e. assume there exists a finite set E such that g(x) = f(x)

for all x € [a, b]\ E. Prove that g is integrable on [a, b] and that få g(x) dx = f f(x)dx. Hint:

think about applying induction on your result from part (b).