Math 125A: Topics for Midterm 1

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The midterm will cover Chapter 5, with appropriate background material from Chapter 4, in the text by Thomson, Bruckner, and Bruckner. There won't be questions specifically about the topics in Chapter 4, but you need to know about open sets, closed sets, compact sets, neighborhoods, accumulation points etc. to answer questions about limits and continuous functions.

We skipped a few of the "enrichment sections" in the text, and the midterm will be based on the material that we did cover in class or in homeworks. Here's a detailed list of the corresponding sections. Disclaimer: The midterm may still ask you to figure out things you haven't seen before and that aren't listed explicitly here!

Chapter 4. Sets of Real Numbers

• 4.2 Points
  1. 4.2.1 Interior Points
  2. 4.2.2 Isolated Points
  3. 4.2.3 Points of Accumulation
• 4.3 Sets
  1. 4.3.1 Closed Sets
  2. 4.3.2 Open Sets
• 4.4 Elementary Topology
• 4.5 Compactness Arguments
  1. 4.5.1 Bolzano-Weierstrass Property
  2. 4.5.5 Compact Sets

Chapter 5. Continuous Functions

• 5.1 Introduction to Limits
  1. 5.1.1 Limits (ε-δ Definition)
  2. 5.1.2 Limits (Sequential Definition)
  3. 5.1.3 Limits (Mapping Definition)
  4. 5.1.4 One-Sided Limits
  5. 5.1.5 Infinite Limits
• 5.2 Properties of limits
  1. 5.2.1 Uniqueness of Limits
  2. 5.2.2 Boundedness of Limits
  3. 5.2.3 Algebra of Limits
  4. 5.2.4 Order Properties
  5. 5.2.6 Examples
• 5.4 Continuity
  1. 5.4.1 How to Define Continuity
  2. 5.4.2 Continuity at a Point
  3. 5.4.3 Continuity at an Arbitrary Point
  4. 5.4.4 Continuity on a Set
• 5.5 Properties of Continuous Functions
• 5.6 Uniform Continuity
• 5.7 Extremal Properties
• 5.8 Darboux Property
• 5.9 Points of Discontinuity
  1. 5.9.1 Types of Discontinuities
  2. 5.9.2 Monotonic Functions