Analysis Exercises

Real Analysis

  • Let C be the Cantor set. Let f be a bounded real function on [0,1] which is continuous at every point outside of C. Prove that f \in \mathcal{R} on [0,1].
  • If f(x) = 0 for all irrational x, f(x) = 1 for all rational x, prove that f \notin \mathcal{R} on [a,b] for any a < b.
  • Let I = [0,1] be the closed unit interval. Suppose f is a continuous mapping of I into I. Prove that f(x) = x for at least one x \in I.
  • If f is defined on E, the graph of f is the set of points (x,f(x)), for x \in E. In particular, if E is the set of real numbers, and f is real-valued, the graph of f is a subset of the plane. Suppose E is compact, and prove that f is continuous if and only if its graph is compact.
  • Associate to each sequence a = {\alpha_{n}}, in which \alpha_{n} is 0 or 2, the real number x(a) = \sum_{n = 1}^{\infty}\frac{\alpha_{n}}{3^{n}}. Prove that the set of all x(a) is precisely the Cantor set.
  • Suppose f(x)f(y) = f(x+y) for all real x and y. Assuming that f is differentiable and not zero, prove that f(x) = e^{cx}. Prove the same thing, assuming only that f is continuous.
  • A function f:\mathbb{R} \to \mathbb{R} is said to have infinite derivative everywhere if for all x \in \mathbb{R}, \lim\inf_{y \to x}(f(x)-f(y))/(x-y) = +\infty. Show that no functions have infinite derivative everywhere.
  • Find a sequence of functions {f_{n}:\mathbb{R} \to \mathbb{R}} converging to 0 almost everywhere but not converging to 0 in L^{1}.
  • Find a sequence of function {f_{n}:\mathbb{R} \to \mathbb{R}} converging to 0 pointwise but not converging to 0 in L^{1}.
  • Let f:\mathbb{R} \to \mathbb{R} be any function such that if x is in the domain then so is 1/x and f(x)+f(1/x) = x. What is the largest domain of all such functions?
  • Let A be a closed set of \mathbb{R}. Show that there exists a continuous function f:\mathbb{R} \to \mathbb{R} such that f(x) = 0 if and only if x \in A.
  • Let f:[0,1] \to [0,1] be a continuous function. Prove that the arc length of f is well-defined, 2 is an upper bound for the arc length, and give an example of a well-known continuous function f with arc length 2.

Complex Analysis

  • Write three terms of the Laurent expansion of f(x) = \frac{1}{z(z-1)(z+1)} in the annulus 1 < |z|.
  • Show that \sum_{n = 1}^{\infty}\frac{z^{n}}{n} converges at all points on the unit circle |z| = 1 except z =1.
  • Show that a holomorphic function f with |f(z)| = 1 for all z is constant.
  • Let f be holomorphic and bounded on the upper half-plane \mathbb{H} and takes real values on \mathbb{R}. Prove f is constant.
  • Evaulate \int_{-\infty}^{\infty}\frac{e^{ix}}{1+x^{2}}\,dx.
  • Prove that a function which is analytic in the whole plane and satisfies an inequality |f(z)| < |z|^{n} for some n and all sufficiently large |z| reduces to a polynomial.
  • Show that a function which is analytic in the whole plane and has a nonessential singularity at \infty reduces to a polynomial. Use this to show that the functions e^{z}, \sin{z} and \cos{z} have essential singularities at \infty.
  • How many roots does the equation z^{7}-2z^{5}+6z^{3}-z+1 = 0 have in the disk |z| < 1?
  • Find the poles and residues of \frac{1}{\sin^{2}{z}}.
  • Evaluate the integral \int_{0}^{\infty}\frac{x^{2}}{(x^{2}+a^{2})^{3}}\,dx by the method of residues.

Fourier Analysis

  • Let f \in C^{1}(\mathbb{T}). Prove that \sum_{n = -\infty}^{\infty}|\hat{f}(n)|n^{2} < \infty.
  • Use the Fourier coefficients of the Sawtooth function s(x) = x-\pi on [0,2\pi] to prove \sum_{n = 1}^{\infty}\frac{1}{n^{2}} = \frac{\pi^{2}}{6}.
  • For any measurable E \subset [0,1], show that \lim_{n \to \infty}\int_{E}e^{2\pi inx}\,dx = 0.