Homework 6 Math 118B, Winter 2010
Due on Thursday, March 11th, 2010
1. Let
0
sin2
fn (x) =
0
x
1
n+1
1
x < n+1 ,
1
x n,
1
x > n.
Show that cfw_fn converges to a continuous function, but not uniformly.
Use the series
fn to show that absolute convergence,
Homework 5 Math 118B, Winter 2010
Due on Thursday, March 4th, 2010
1. Show that if f 0 and if f is monotonically decreasing, and if
n
n
cn =
k=1
f (k )
f (x) dx,
1
then
lim cn
n
exists.
2. Let n (x) be positive-valued and continuous for all x [1, 1], wit
Homework 3 Math 118B, Winter 2010
Due on Thursday, February 4, 2010
1. Let be a xed increasing function on [a, b]. For u R(), dene
1/2
b
u
2
2
=
|u| d
.
a
Suppose f, g, h R(), and prove the triangle inequality
f h
2
f g
2
+ g h 2,
as a consequence of the
Homework 2 Math 118B, Winter 2010
Due on Thursday, January 28, 2010
1. Suppose increases on [a, b], a x0 b, is continuous at x0 , f (x0 ) =
1, and f (x) = 0 if x = x0 . Prove that f R() and that f d = 0.
b
a
2. Suppose that f 0, f is continuous on [a, b],
Homework 1 Math 118B, Winter 2010
Due on Thursday, January 14, 2010
1. Let f be dened for all real x, and suppose that M > 0 and > 0
such that
|f (x) f (y )| M |x y |1+ , x, y R.
Prove that f is constant.
2. Suppose that f (x) > 0 in (a, b). Prove that f
Homework 1 Math 118B, Winter 2010
Due on Thursday, January 14, 2010
1. Let f be dened for all real x, and suppose that M > 0 and > 0
such that
|f (x) f (y )| M |x y |1+ , x, y R.
Prove that f is constant.
Proof: If we divide by |x y |, we get
f (x) f (y )