Mathematics Homework Help

5. Suppose f and g are dierentiable on (a; b) and f0(x) = g0(x) for all x 2 (a; b). Show

that f(x) = g(x) + c for some c 2 R.

6. Let f : [a; b] ! R and suppose there exist > 0 and M > 0 such that

jf(x) f(y)j Mjx yj;

for all x; y 2 [a; b].

(a) Show that f is uniformly continuous on [a; b].

(b) Suppose that > 1. Show that f must be constant.

9. (a) Let f : [a; b] ! R be bounded. Prove that f is Riemann integrable on [a; b] if and

only if there is a sequence of partitions fPng1 n=1 such that

lim

n!1

U(f;Pn) L(f;Pn)

= 0:

(b) For each n, let Pn be the partition of [0; 1] into n equal sub-intervals. Find formulas

for U(f;Pn) and L(f;Pn) if f(x) = x.

(c) Use part (a) to show that f(x) = x is Riemann integrable on [0; 1]. What is

1

0 xdx?