Sample questions
1.
(a)
Complete the following definition:
Let f be a real-valued function defined on a set
. f is uniformly continuous
on S if ...
(b)
Prove that
(c)
2.
on
satisfies the definition of uniform continuity.
Give an example of a function which is continuous, but not uniformly continuous. Prove that
your function does not satisfy the definition of uniform continuity.
(a)
Prove that if
(b)
is a convergent series with
and
converges.
Is the theorem true if we remove the restriction that
3.
, then
? If true, prove it. If not, give a
counterexample.
(a)
Prove that if
and
(
), then
.
(b)
Prove that if
and
, then
.
(c)
Prove that if
4.
(a)
(b)
and
, then
.
State the Mean Value Theorem.
Determine whether the Mean Value Theorem holds for the following functions on the specified
intervals. If the conclusion holds, give an example of an interior point which satisfies the
theorem. If the conclusion fails, state which hypothesis of the Mean Value Theorem fails.
i.
ii.
on [-1,2]
on
iii.
on [-1,2]
iv.
on [-1,1]
v.
on [1,3]
vi.
on [-2,3], where
(c)
Prove that
5.
(a)
(b)
for all
.
State Taylor's remainder theorem.
Use Taylor's remainder theorem to prove that the Taylor series for
6.
converges at x=1.
(a)
(b)
Prove the Bolzano-Weierstrass Theorem: Every bounded sequence of real numbers has a
convergent subsequence.
Consider the sequence (x ) defined recursively as follows:
n
Show that (x ) converges and find the limit.
n
(c)
Investigate the convergence of (x ), given
n
7.
(a)
(b)
State the definition of a Cauchy sequence and show that every convergent sequence is a
Cauchy sequence.
Show directly (from the definition) that if
then (x ) is not a Cauchy sequence.
n
(c)
Show directly (from the definition) that if
n-1
then (x ) is a Cauchy sequence. (Hint: first show that n! < 2
n
8.
.)
(a)
Prove that `` Let
continuity'' implies sequential continuity, i.e.,
. Suppose that for any
satisfies
there exists a
, then
with
such that if
. Prove that for any sequence
, we have
.
(b)
Let
be defined by
Prove that f is continuous at x=1/2 and discontinuous everywhere else.
9.
(a)
Define: ``uniform convergence'' of a sequence of functions (f ) defined on a set D.
n
(b)
Prove that if f is continuous on D
n
(c)
and
uniformly on D, then f is continuous
on D.
Give an example of a sequence of continuous functions f on a set D such that the pointwise
n
limit
10.
(a)
(b)
is defined on D, but f is NOT continuous on D.
State the definition of the Riemann integral of a bounded function f over an interval [a,b].
Prove that any continuous function f is Riemann integrable on [a,b]. (Your proof should use the
notion of uniform continuity.)
OR
(c)
Prove that if f is monotone increasing on [a,b], then
(d)
Show that
exists.
by interpreting the sums as Riemann sums for the definite integral of some continuous
function over [0,1].
11.
(a)
Let
i.
ii.
iii.
(b)
For what values of a is f differentiable at x=0?
For what values of a is f continuous at x=0?
When f is differentiable at x=0, does f''(0) exist?
Let
12.
be defined by the property
Suppose that f is continuous at zero. Show that f must be continuous everywhere.
(a)
Find the radius of convergence of each power series:
i.
ii.
iii.
(b)
Show that
Maclaurin series for
13.
(a)
and state the region of validity. (Hint: Start with the
.)
Examine each series for convergence/divergence:
i.
ii.
(b)
Prove that the series
converges if p>2 and diverges if
.