Midterm II Review for Math 421, Fall 2009

Technical terms you must know.

You must be able to give a careful, complete and accurate of these terms. You must also know examples of all of these terms (and their negations or combinations if that makes sense).
  1. an open or closed set
  2. a limit point of a set
  3. the closure of a set
  4. a compact set
  5. an open cover of a set
  6. a connected set
  7. a functional limit (both ε-δ and topological forms)
  8. a continuous function
  9. a uniformly continuous function
  10. the derivative of a function, a differentiable function

Theorems/Results/Axioms/Examples you must know.

This means you must be able to state them and to give examples of them being true and examples of how the conclusion is false if individual hypothesese are dropped (for the theorems), and discuss why (i.e., prove, albeit with a reduced standard of rigor) the claimed properties hold true (for the examples).
  1. results relating different definitions of "closed set"
  2. results on finite&arbitrary unions&intersections of open&closed sets
  3. the Heine-Borel Theorem
  4. results giving different characterizations of compactness
  5. examples of functions with interesting discontinuity sets:
  6. the Divergence Criterion for Functional Limits
  7. the theorem giving four different characterizations of continuity
  8. theorems on continuous functions preserving compactness and connectedness (also the EVT and IVT versions)
  9. a theorem on [uniformly] continuous functions on compact sets
  10. theorems relating continuity and differentiability
  11. the Interior Extremum Theorem
  12. Darboux's Theorem
  13. Rolle's Theorem
  14. the Mean Value Theorem
  15. L'Hospital's Rule
  16. a continuous function which is non-differentiable at a point, or is non-differentiable everywhere

Theorems you should be able to prove during an in-class midterm:

  1. Heine-Borel
  2. the theorems on continuous functions preserving compactness and connectedness
  3. differentiability implies continuity
  4. the Interior Extremum Theorem
  5. Rolle's Theorem
  6. Some simple propositions/theorems from the book, like: