Math 118-B Winter 2023
1. From Rudin’s book: Chapter 5 exercises 3, 4, 5, 6, 17, 22.
2. From Rudin’s book: Chapter 6 exercises 2, 5, 8, 13, 15.
3. From Rudin’s book: Chapter 7 exercises 2, 5, 7, 15, 18.
4. From Rudin’s book: Chapter 9 exercises 5, 6, 7, 8, 9, 10, 11, 12.
5. Let f : [0, 1] → R be a differentiable function (i.e. f, f ′ are defined
for all x ∈ [0, 1]). Suppose that f ′(x) ≥M > 0, ∀x ∈ [0, 1]. Show that
there is an interval of length 1/4 on which |f | ≥M/4.
6. Let f, g : [a, b]→ R be continuous functions with g(x) ≥ 0, x ∈ [a, b].
(a) Prove that
F (x) =
∫ x
a
f(t)dt, x ∈ [a, b] is differentiable and F ′(x) = f(x).
(b) Prove that there exists c ∈ (a, b) such that
f(c) =
1
b− a
∫ b
a
f(t)dt.
(c) Prove that there exists d ∈ (a, b) such that
f(d)
∫ b
a
g(x)dx =
∫ b
a
f(x)g(x)dx.
7. Let f : R→ R be defined as f(x) = e−|x|.
(a) Prove that f ′(x) is defined everywhere except at x = 0.
(b) Prove that f is uniformly continuous in R.
8. Let f : [0, 1]→ R be a continuous function.
(a) Prove (using just the definition) that f is Riemann integrable.
(b) Prove that
lim
n→∞
1
n
n∑
k=1
f
(
k
n
)
=
∫ 1
0
f(t)dt.
1
29. Prove that if f : [a, b]→ R is continuous, then
∃ ξ ∈ [a, b] s. t
∫ b
a
f(x)dx = (b− a)f(ξ). (1)
(b) Give an example of f : [a, b] → R Riemann integrable for which
(??) does not hold.
10. Prove that if
a0
1
+
a1
2
+ ...+
an
n+ 1
= 0,
then
P (x) = a0 + a1x+ ...+ anx
n = 0, for some x ∈ [0, 1].
11. Let f : [0, 1]→ R be a differentiable function on [0, 1], such that
f(0) = 0, and ∃A > 0 s.t. ∀x ∈ [0, 1] : |f ′(x)| ≤ A |f(x)|.
Prove that f(x) = 0, ∀x ∈ [0, 1].
12. Let f : (a, b)→ R be a C2((a, b)) function, i.e. f ′(x), f ′′(x) defined in
(a, b) with f ′′ continuous on (a, b). Suppose f(x0) = f ′(x0) = f ′′(x0) =
0 for some x0 ∈ (a, b). Show that the function g : (a, b) → R with
g(x) = f(x)/(x− x0)2 is continuous on (a, b).
13. Prove that the function f : R→ R defined as
f(x) =
{
e−1/x, x > 0,
0, x ≤ 0.
has derivatives of all order at x = 0.
14. Let f : [0, 1]→ R be defined as
f(x) =

2k−1 − 1
2k−1
, x ∈ [2
k−1 − 1
2k−1
,
2k − 1
2k
), k = 1, 2, , .....
1, x = 1.
Draw a graph of f . Where is f discontinuous ? Prove that f is Riemann
integrable on [0, 1].
315. Let g : [0, 1] → R be a continuous function with g(0) = 0. Prove that
the sequence fn(x) = x
ng(x) converges uniformly on [0, 1].
16. Is the sequence {sin(nx) : n ∈ N} equicontinuous on [0, 1]?
17. Let gn : [0, 1]→ R be a sequence of twice differentiable functions such
that for n ∈ N, gn(0) = g′n(0) = 0. Assuming that the {g′′n : n ∈ N}
are uniformly bounded prove that there is subsequence of {gn : n ∈ N}
which converges uniformly.
18. Let
f(x) =
∞∑
k=1
cos(nx)
n4
.
Find a series representation for the function∫ x
0
f(t)dt.
19. Let f : [0, 1]→ R be continuously differentiable, with f(0) = 0. Prove
that
‖f‖2∞ ≤
∫ 1
0
(f ′(x))2dx.
20. Let fn : [0, 1] → R be a sequence of differentiable functions, such that
fn(0) = 0, ∀n ∈ N and {f ′n : n ∈ N} are uniformly bounded. Suppose
that fn(x) converges to g(x) for each x ∈ [0, 1] (pointwise convergence).
Prove that g is continuos.
fn(x) =
nx
nx+ 1
, n = 1, 2, ....
(a) Find the point-wise limit of {fn} on [0, 1]. Is the convergence uni-
form?
(b) Find
lim
n→∞
∫ 1
0
fn(x)dx,
∫ 1
0
lim
n→∞
fn(x)dx.
21. Same as in (2) for fn(x) = nx e
−nx2 .
422. Let fn : R → R be a sequence of functions such that f ′(x), f ′′(x) are
defined for all x ∈ R. Assuming that fn → 0 uniformly on R, and that
the sequence f ′′n is uniformly bounded in R, prove that
f ′n → 0 uniformly.
23. Compute
∞∑
n=1
n2
3n
,
∞∑
n=1
1
n2 2n
24. Let k ∈ N. Define the sequence fn : R→ R as
fn(x) =
xk
x2 + n
.
For which values of k does the sequence converges uniformly on R?
25. Let gn : [0, 1] → R be a sequence of uniformly bounded Riemman
integrable functions. Define
Gn(x) =
∫ x
0
gn(t)dt.
prove that a subsequence of {Gn : n ∈ N} converges uniformly.
26. Let (X, d) be a COMPLETE metric space. Let φ : X → X be a
contraction, i.e.
∃ θ ∈ (0, 1) ∀x, y ∈ X d(φ(x), φ(y)) ≤ θ d(x, y).
Prove that there exists a unique x∗ ∈ X such that φ(x∗) = x∗ (fixed
point).
HINT. Let x0 ∈ X any point and x1 = φ(x0) and xn+1 = φ(xn), n ∈ N.
Prove :
d(xn, xn+k) ≤ θk d(x0, x1).
Prove that (xn)
∞
n=1 is a Cauchy sequence.
527. Let f : R→ R be acontinuous function such that∫ ∞
−∞
|f(x)|dx <∞.
Show that there exists a sequence {xn : n ∈ N} such that
xn →∞, xn f(xn)→ 0, xnf(−xn)→ 0 as n ↑ ∞.
28. Let Pn be a sequence of polynomials of degree ≤ 10. Suppose that
the sequence Pn(x) convergences (pointwise) to zero for each x ∈ [0, 1].
Prove that the sequence Pn convergences uniformly to zero.
29. State the Arzela-Ascoli Theorem. Give examples showing that ech of
the three main hypotheses are necessary.
30. State the Weierstrass Approximation Theorem.