Mathematics Homework Help
UCLA Darboux Theorem Intermediate Value Property & Derivatives Exercise Examples
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?
