Analysis Homework #6
due Thursday, Feb. 17
1. Suppose
∫ 1 f is non-negative and integrable on [0, 1] with f (x) = 0 for all x ∈ Q. Show
that 0 f (x) dx = 0.
2. Suppose f is increasing on [0, 1]. Show that f is integrable on [0, 1]. Hint: let P be
the partition of [0, 1] into n subintervals of the same length and show that
S + (f, P ) − S − (f, P ) =
f (1) − f (0)
.
n
3. Show that there exists a unique function f , defined for all x ∈ R, such that
f ′ (x) =
1
,
1 + x4
f (0) = 0.
4. Suppose f is continuous on [a, b]. Show that there exists some c ∈ (a, b) such that
∫
b
f (t) dt = (b − a) · f (c).
a
Hint: apply the mean value theorem to the function F (x) =
∫x
a
f (t) dt.
• You are going to work on these problems during your Thursday tutorial (at 4 pm).
• When writing up solutions, write legibly and coherently. Use words, not just symbols.
• Write your name and then MATHS/TP/TSM on the first page of your homework.
• Your solutions may use any of the results stated in class (but nothing else).
• NO LATE HOMEWORK WILL BE ACCEPTED.