Tags: #active_projects #qual_real_analysis
Week 1: Preliminaries
Topics

Continuity and uniform continuity
 Pathological functions and sequences of functions

Convergence
 The Cauchy criterion
 Uniform convergence
 The \(M\)Test
 HeineBorel
 Normed spaces

Series and sequences,
 Convergence
 Small tails,
 limsup and liminf,
 Cauchy criteria for sums and integrals
 Variation and bounded variation
Things that don’t explicitly appear in qual problems
 Baire category theorem,
 Nowhere density,
 Basic inequalities (triangle, CauchySchwarz)

Concepts from Calculus
 Mean value theorem
 Taylor expansion
 Taylor’s remainder theorem
 Intermediate value theorem
 Extreme value theorem
 Rolle’s theorem
 Riemann integrability
 Weierstrass approximation
Background / Warmup / Review
 Derive the reverse triangle inequality from the triangle inequality.
 Let \(E\subseteq {\mathbf{R}}\). Define \(\sup E\) and \(\inf E\).
 What is the Archimedean property?
Metric Spaces / Topology
 What does it mean for a metric space to be complete?
 Give two or more equivalently definitions for compactness in a complete metric space.
 What is an interior point? An isolated point? A limit point?
 What does it mean for a set to be open? Closed?
 What is the closure of a subspace \(E\subseteq X\)?
 What does it mean for \(E\subseteq X\) to be a dense subspace?

What does it mean for a family of sets to form a basis for a topology?
 What is a basis for the standard topology on \({\mathbf{R}}^d\)?

Let \(X\) be a subset of \({\mathbf{R}}^d\). Prove the HeineBorel theorem:
 Show that \(X\) compact \(\implies X\) is closed
 Show that \(X\) compact \(\implies X\) is bounded
 Show that a closed subset of a compact set must be bounded.
 Show that if \(X\) closed and bounded \(\implies X\) is compact.

Find an example of a metric space with a closed and bounded subspace that is not compact.
 How can this be modified to obtain a necessary and sufficient condition?

Determine if the following subsets of \({\mathbf{R}}\) are opened, closed, both, or neither:
 \({\mathbf{Q}}\)
 \({\mathbf{Z}}\)
 \(\left\{{1}\right\}\)
 \(\left\{{p \in {\mathbf{Z}}^{\geq 0} {~\mathrel{\Big\vert}~}p\text{ is prime}}\right\}\)
 \(\left\{{ {1\over n} {~\mathrel{\Big\vert}~}n\in {\mathbf{Z}}^{\geq 0}}\right\}\)
 \(\left\{{ {1\over n} {~\mathrel{\Big\vert}~}n\in {\mathbf{Z}}^{\geq 0}}\right\} \cup\left\{{0}\right\}\)
Sequences
 Can a convergent sequence of real numbers have a subsequence converging to a different limit?

What does it mean for a sequence of functions to converge pointwise and to converge uniformly?
 Give an example of a sequence that converges pointwise but not uniformly.
 Prove that every sequence admits a monotone subsequence.
 Prove the monotone convergence theorem for sequences.
 Prove the BolzanoWeierstrass Theorem.
Series
– What does it mean for a series to converge? How can you check this?
 What does it mean for a series to converge uniformly? What do you have to show to prove it does not converge uniformly?
 Show that if \(\sum_{n\in {\mathbb{N}}} a_n < \infty\) converges, then `
Continuity and Discontinuity

What does it mean for a function to be uniformly continuous on a set?

Is it possible for a function \(f:{\mathbf{R}}\to {\mathbf{R}}\) to be discontinuous precisely on the rationals \({\mathbf{Q}}\)? If so, produce such a function, if not, why?
 Can the set of discontinuities be precisely the irrationals \({\mathbf{R}}\setminus{\mathbf{Q}}\)?

Find a sequence of continuous functions that does not converge uniformly, but still has a pointwise limit that is continuous.
Exercises

Find a function that is differentiable but not continuously differentiable.

Prove the uniform limit theorem: a uniform limit of continuous function is continuous.

Show that the uniform limit of bounded functions is uniformly bounded.

Construct sequences of functions \(\left\{{f_n}\right\}_{n\in {\mathbb{N}}}\) and \(\left\{{g_n}\right\}_{n\in {\mathbb{N}}}\) which converge uniformly on some set \(E\), and yet their product sequence \(\left\{{h_n}\right\}_{n\in {\mathbb{N}}}\) with \(h_n \coloneqq f_n g_n\) does not converge uniformly.
 Show that if \(f_n, g_n\) are additionally bounded, then \(h_n\) does converge uniformly.

Find a sequence of functions such that \begin{align*}\frac{d}{d x} \lim _{n \rightarrow \infty} f_{n}(x) \neq \lim _{n \rightarrow \infty} \frac{d}{d x} f_{n}(x)\end{align*}

Find a uniform limit of differentiable functions that is not differentiable.

Prove that the Cantor set is a Borel set.

Show the Cantor ternary set is totally disconnected; that is show it contains no nonempty open interval.


Qual Questions

 Note: outline omitted!