Assignment 2 of Math 5302
1. Complete the proof of Theorem 2.3 in the lecture notes by showing that a decreasing function on
[a, b] is integrable.
2. Let f be a bounded function on [a, b], so that there exists B > 0 such that |f(x)| ≤ B for all
x ∈ [a, b].
(a) Show
U(f
2
, P) − L(f
2
, P) ≤ 2B[U(f, P) − L(f, P)]
for all partitions P of [a, b]. Hint: f
2
(x) − f
2
(y) = (f(x) + f(y))(f(x) − f(y)).
(b) Show that if f is integrable on [a, b], then f
2
is also integrable on [a, b].
3. Let f be a bounded function on [a, b]. Suppose that f
2
is integrable on [a, b]. Must f also be
integrable on [a, b]?
4. Suppose that f and g are integrable on [a, b]. Show that max(f, g) is also integrable on [a, b].
Hint: Derive and apply the formula
max(f, g) = 1
2
(f + g + |f − g|).
5. Suppose f and g are continuous functions on [a, b] such that R b
a
f =
R b
a
g. Prove there exists x in
(a, b) such that f(x) = g(x).
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