Homework 9
Math 501
Due November 7, 2014
Exercise 1
Let f : [a, b] → R be a function. A function F is called an antiderivative
of f if F 0 (x) = f (x) for all x ∈ R. Prove that two antiderivatives of f must
differ by a constant.
Exercise 2
Let f : [a, b] → R be Riemann integrable and let Z ⊂ [a, b] be a zero set.
Suppose that f (x) = 0 for all x ∈ [a, b] \ Z. Prove that
Z b
f dx = 0.
a
Exercise 3
Let f, g : [0, 1] → R be Riemann integrable functions. Let C be the middlethirds Cantor set. Suppose that f (x) = g(x) for all x ∈ [0, 1] \ C. Prove that
Z 1
Z 1
f dx =
g dx
0
0
Exercise 4
Let f, g : [a, b] → R be Riemann integrable. Prove that if f (x) ≤ g(x) for all
x ∈ [a, b], then
Z b
Z b
f dx ≤
g dx.
a
a
Remark. If you want to be ambitious, try to prove the result even holds when
we loosen the hypothesis to f (x) ≤ g(x) for all x ∈ [a, b] \ Z where Z ⊂ [a, b] is
any zero set.
1
Exercise 5
Let f : [a, b] → R be a bounded function and let α : [a, b] → R be increasing
and differentiable. Suppose that α0 is Riemann integrable. Prove that f is
Riemann-Stieltjes integrable with respect to α if and only if f α0 is Riemann
integrable. When this is the case, show that
b
Z
Z
f dα =
a
b
f α0 dx.
a
Remark. Note that if P = {x0 , . . . , xn } is a partition, by the mean value theorem, there exist ti ∈ (xi−1 , xi ) for each i = 1, . . . , n such that
∆αi = α0 (ti )∆xi .
2