Real Analysis
- Let
be the Cantor set. Let
be a bounded real function on
which is continuous at every point outside of
. Prove that
on
.
- If
for all irrational
,
for all rational
, prove that
on
for any
.
- Let
be the closed unit interval. Suppose
is a continuous mapping of
into
. Prove that
for at least one
.
- If
is defined on
, the graph of
is the set of points
, for
. In particular, if
is the set of real numbers, and
is real-valued, the graph of
is a subset of the plane. Suppose
is compact, and prove that
is continuous if and only if its graph is compact.
- Associate to each sequence
, in which
is
or
, the real number
. Prove that the set of all
is precisely the Cantor set.
- Suppose
for all real
and
. Assuming that
is differentiable and not zero, prove that
. Prove the same thing, assuming only that
is continuous.
- A function
is said to have infinite derivative everywhere if for all
,
. Show that no functions have infinite derivative everywhere.
- Find a sequence of functions
converging to
almost everywhere but not converging to
in
.
- Find a sequence of function
converging to
pointwise but not converging to
in
.
- Let
be any function such that if
is in the domain then so is
and
. What is the largest domain of all such functions?
- Let
be a closed set of
. Show that there exists a continuous function
such that
if and only if
.
- Let
be a continuous function. Prove that the arc length of
is well-defined,
is an upper bound for the arc length, and give an example of a well-known continuous function
with arc length
.
Complex Analysis
- Write three terms of the Laurent expansion of
in the annulus
.
- Show that
converges at all points on the unit circle
except
.
- Show that a holomorphic function
with
for all
is constant.
- Let
be holomorphic and bounded on the upper half-plane
and takes real values on
. Prove
is constant.
- Evaulate
.
- Prove that a function which is analytic in the whole plane and satisfies an inequality
for some
and all sufficiently large
reduces to a polynomial.
- Show that a function which is analytic in the whole plane and has a nonessential singularity at
reduces to a polynomial. Use this to show that the functions
,
and
have essential singularities at
.
- How many roots does the equation
have in the disk
?
- Find the poles and residues of
.
- Evaluate the integral
by the method of residues.
Fourier Analysis
- Let
. Prove that
.
- Use the Fourier coefficients of the Sawtooth function
on
to prove
.
- For any measurable
, show that
.