Math 447, Homework 1.∗
1. As discussed on page 203, partitions are ordered by refinement P ⊂ Q.
(a) Prove that any two partitions P and Q have a common refinement: a partition R such that
P ⊂ R and Q ⊂ R. (Very easy!)
(b) Define V = limP V (f, P ) is for any ε > 0, there is a partition P0 such that for any P ⊃ P0 ,
|V (f, P ) − V | < ε. Prove that
lim V (f, P ) = Vab f,
P
including the statement that if one of them is finite, so is the other.
2.
(a) Let
a = a0 < a1 < a2 < . . . < an = b.
Suppose f is monotone on every interval [ai−1 , ai ]. Prove that
n
X
Vab f =
|f (ai ) − f (ai−1 )| .
i=1
(Hint: do not do this directly from the definition...)
(b) Let α ∈ R and f (x) = xα cos(π/x). Show that f ∈ BV [0, 1] if and only if α > 1. (Cf.
Exercise 6.)
3. Exercise 13.7 (page 205).
4. Exercise 13.9 (page 205). See Exercise 19 for a possible hint, although you can prove this result
without it.
Quiz 1. Exercises 13.10 and 13.11 (page 205).
∗
c 2016 by Michael Anshelevich.
1
Math 446, Honors Homework 1
1. As discussed on page 203, partitions are ordered by refinement P ⊂ Q.
(a) Prove that any two partitions P and Q have a common refinement: a partition R such that
P ⊂ R and Q ⊂ R. (Very easy!)
(b) Define V = limP V (f, P ) is for any ε > 0, there is a partition P0 such that for any P ⊃ P0 ,
|V (f, P ) − V | < ε. Prove that
lim V (f, P ) = Vab f,
P
including the statement that if one of them is finite, so is the other.
2.
(a) Let
a = a0 < a1 < a2 < . . . < an = b.
Suppose f is monotone on every interval [ai−1 , ai ]. Prove that
n
X
Vab f =
|f (ai ) − f (ai−1 )| .
i=1
(Hint: do not do this directly from the definition...)
(b) Let α ∈ R and f (x) = xα cos(π/x). Show that f ∈ BV [0, 1] if and only if α > 1. (Cf.
Exercise 6.)
3. Exercise 13.7 (page 205).
4. Exercise 13.19 (page 210). Note a misprint in part (c).
Honors Quiz 1. Exercises 13.10 and 13.12 (page 205).