MAA 4212, Spring 2009—Homework # 1 non-book problems
Hand in only A2.
A1. Let a, b ∈R, a < b, and let f: [a, b]→R. Prove that if fis integrable on [a, b], then
for any sequence (Pn, Tn) of pointed partitions for which kPnk → 0 as n→ ∞,
lim
n→∞
S(f;Pn, Tn) = Zb
a
f.
(Hence the integral can be evaluated by taking such a limit, if you know ahead of time
that fis integrable.)
A2 (formerly called A1). For any real-valued function f, the positive part of f, denoted
f+, and negative part of f, denoted f−, are defined by f+(x) = max{f(x),0}and f−(x) =
−min{f(x),0}. (Thus both f+and f−are non-negative, and f=f+−f−[why?].)
Let a, b ∈R, a < b, and let f: [a, b]→R. Prove that if fis integrable on [a, b], then
so are f+and f−.
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