Introduction to Analysis
Homework 8
1. (Rudin ex6.3) Define three functions β1, β2, β3 as followsβj(x) = 0 if x <0, βj(x) = 1 if x > 0 for j = 1,2,3; and β1(0) = 0, β2(0) = 1, β3(0) = 12. Let f be a bounded function on [−1,1].
(a) Prove thatf ∈ R(β1) if and only iff(0+) =f(0) and that then Z
f dβ1 =f(0).
(b) State and prove a similar result forβ2.
(c) Prove thatf ∈ R(β3) if and only iff is continuous at 0.
(d) Iff is continuous at 0 prove that Z
f dβ1 = Z
f dβ2 = Z
f dβ3 =f(0).
2. (Rudin ex6.5) Supposef is a bounded real function on [a, b], and f2 ∈ R on [a, b]. Does it follow that f ∈ R? Does the answer change if we assume that f3 ∈ R?
3. (Rudin ex6.8) Suppose f ∈ R on [a, b] for every b > a where a is fixed. Define Z ∞
a
f(x)dx= lim
b→∞
Z b
a
f(x)dx
if this limit exists (and is finite). In that case , we say that the integral on the left converges. If it also converges after f has been replaced by |f|, it is said to converge absolutely.
Assume that f(x)≥0 and that f decreases monotonically on [1,∞). Prove that Z ∞
1
f(x)dx converges if and only if
∞
X
n=1
f(n)
converges.
4. (Rudin ex6.10) Letp and q be positive real numbers such that 1
p+ 1 q = 1.
Prove the following statements.
(a) Ifu≥0 and v ≥0, then
uv ≤ up p +vq
q . Equality holds if and only if up =vq.
(b) Iff ∈ R(α), g ∈ R(α), f ≥0, g ≥0, and Z b
a
fpdα= 1 = Z b
a
gqdα, then
Z b
a
f gdα≤1.
(c) Iff and g are real functions inR(α), then
Z b
a
f gdα
≤nZ b a
|f|pdα
o1/pnZ b
a
|g|qdα o1/q
.
This isH¨older’s inequality. Whenp=q= 2 it is usually called the Schwarz inequal- ity.
5. Prove that f ∈ R(α) if f is increasing and is continuous at every point where α is discontinuous.
6. If P = {x0,· · · , xn} is a partition of [a, b], let k P k= sup4xi. In some texts, one sees the Riemann integral defined as a limit over partitions withkP k→0. This exercise will show that such a definition is equivalent to the one given in Rudin.
(a) Suppose f is a function on [a, b] and M a constant such that |f(x)| ≤ M for all x. Show that if P is a partition of [a, b], and P∗ a refinement obtained by adding a single point to P, then U(P, f)− U(P∗, f) ≤ 2M k P k, and similarly that L(P∗, f)−L(P, f)≤2M kP k.
(b) Show that the following conditions on a bounded functionf on [a, b] are equivalent:
(i) f ∈ R.
(ii) For every > 0, there exists a δ > 0 such that for every partition P of [a, b]
with kP k< δ, one has U(P, f)−L(P, f)< . 7. (a) Letf : [0,1]→R be defined by
f(x) =
0, xis irrational
1
n, x= mn in lowest terms.
Show that for every continuous increasing function α on [0,1] one has f ∈ R(α), and R1
0 f dα= 0. (You may do this by doing part (b) below, if your choose.)
(b) Show that the same conclusion is true of any functionf on [0,1] with the property that for every >0, the set{x∈[0,1]
|f(x)| ≥}is finite. Show that the function f of part (a) has that property.
8. Show by example that, iff andφ are Riemann-integrable functions, their compositeφ◦f need not be Riemann-integrable. (Hint: As the discuss in class, we know that if f ∈ R and φis continuous, thenφ◦f ∈ R. Consider the functionf is defined in Problem 7 and construct a discontinuous function φ.)
