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 .