Tags: #active_projects #qual_real_analysis
Week 1: Preliminaries
Topics
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Continuity and uniform continuity
- Pathological functions and sequences of functions
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Convergence
- The Cauchy criterion
- Uniform convergence
- The M-Test
- Heine-Borel
- Normed spaces
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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, Cauchy-Schwarz)
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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⊆R. Define sup 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?
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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?
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Let X be a subset of {\mathbf{R}}^d. Prove the Heine-Borel 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.
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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?
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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?
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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 Bolzano-Weierstrass 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
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What does it mean for a function to be uniformly continuous on a set?
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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}}?
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Find a sequence of continuous functions that does not converge uniformly, but still has a pointwise limit that is continuous.
Exercises
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Find a function that is differentiable but not continuously differentiable.
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Prove the uniform limit theorem: a uniform limit of continuous function is continuous.
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Show that the uniform limit of bounded functions is uniformly bounded.
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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.
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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*}
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Find a uniform limit of differentiable functions that is not differentiable.
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Prove that the Cantor set is a Borel set.
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Show the Cantor ternary set is totally disconnected; that is show it contains no nonempty open interval.
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Qual Questions
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- Note: outline omitted!