MATH 321 - PRACTICE EXAM #1
The first five problems are similar in length and difficulty to the problems in the actual
exam. Problem 6 is more difficult.
P ROBLEM 1. In each case below determine if the sequence or sum converges pointwise/
uniformly/neither in the given domain:
(a)
1
fn (x) = (1 − x) n , on [0, 1).
(b)
fn (x) = cosn x,
(c)
on [0, 2π].
X sin(nx)
√
,
n3 + 1
on R.
P ROBLEM 2. Let f(x) be bounded and α(x) non-decreasin on [a, b].
Rb
(a) Give a definition of a fdα.
(b) Prove that if a < c < b then
Zb
Zc
Zb
fdα = fdα + fdα.
a
a
c
P ROBLEM 3. Recall that f is uniformly continuous on E if for any ε > 0 there exists a δ > 0,
such that
|f(x) − f(y)| < ε, for any x, y ∈ E,|x − y| < δ.
Prove that if fn converge to f uniformly on E and all fn are uniformly continuous, then f is
also uniformly continuous.
P ROBLEM 4. Let f and g be bounded functions on [a, b], such that f(x) = g(x) for all x
except for a single x0 ∈ [a, b]. Prove that if f ∈ R, then also g ∈ R, and
Zb
Zb
fdx = gdx.
a
a
P ROBLEM 5. Recall the function f[0, 1] → R from the homework:



1 if x = 0,
f(x) =
1
n


0
if x ∈ Q, x =
m
,
n
otherwise.
Does f have bounded variation?
1
m, n ∈ N, gcd(m, n) = 1,
P ROBLEM 6. Let g : [a, b] → R be continuous and have bounded variation. Define
G(x) = Vax g.
Prove that
lim G(x) = 0.
x→a+
Hint: Here are a couple of points to help thinking about this problem:
(a) There exists a continuous function g(x) on [a, b], not of bounded variation, such that
Vax g = ∞ for any x.
(b) Given ε > 0, can you find a partition P of [a, b] such that when we compute the
variation using P, but omit the first piece of P, we get
n
X
|g(xi ) − g(xi−1 )| > Vab g − ε.
i=2
(c) Use the previous part to prove that Vaa+δ g < ε for some δ > 0.