1— M413 — Midterm 3 — Name:
Problem 1 (40 points) Let be bounded on [0, 1]. Assume furthermore that for
every 0 < < 1, is Riemann integrable on the interval [, 1]. Show that is then
also Riemann integrable on [0, 1].
Hints: Let = sup and = inf on [0, 1]. Fix > 0, let < /( −), and
let be a partition of [, 1]. How small can you make ( , ) − ( , )? Define
the partition ′ = {0} ∪ of [0, ]. Now show that ( ′, ) − ( ′, ) can be
made arbitrarily small.
Problem 2 (30 points) Recall that a function : [, ] → ℝ is uniformly continuous
if for every > 0 there exists > 0 so that if , ∈ [, ] with | − | < ,
then | () − ()| < . Assume that is Riemann integrable on [, ], and let
() = ∫

() . Show that is uniformly continuous on [, ].
Problem 3 (30 points) Let : ℝ → ℝ be a continuous function such that () > 0
for all . Define : (−∞,∞) → ℝ by
() =

∫
0
1
()
.
Show first that is invertible. Let = −1 be the inverse function. Now show
that
′() = (()) .

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