Sprecher's Chapters 6-7: Continuity and Differentiability

Chapter 6: Continuity

  1. Limits of functions
    Sequential limits
    Limits defined via neighborhoods
    One-sided limits
  2. Continuous functions
    Definition, via neighborhoods, of continuity at a domain point
    Equivalent characterizations
    Theorem 24.8: A function from set A to R is continuous iff the inverse image of each open set is (relatively) open in A.
    Intermediate Value Theorem
    Continuity of sums, products, and quotients of functions
    Tietze Extension Theorem: We'll look at this and generalize it in Topology.
  3. The nature of discontinuities
    Examples
    First kind: Left- and right-hand limits exist.
    Second kind: At least one does not exist.
    What sets can be sets of discontinuity points of a function? (25.4)
  4. Monotonic functions
    Definition
    Left- and right-hand limits of a monotonic function
    The set of discontinuities of a monotonic function
  5. Uniform continuity
    For a continuous function, consider delta as a function of epsilon at each domain point.
    Definition of uniform continuity
    Theorem 27.4: A continuous function on a compact set is uniformly continuous.
    More theorems

Chapter 7: Differentiability

  1. The derivative at a point
    Definition
    Left and right derivatives
    28.2: Characterization: f is differentiable at c iff f has a linear approximation at c.
    Differentiability of sums, products, quotients, and compositions of differentiable functions
    Examples
  2. A continuous nowhere differentiable function
  3. Properties of the derivative
    Definitions of local maximum and local minimum
    Rolle's theorem
    Generalized Mean Value Theorem
    Mean Value Theorem
    First derivative conditions for a function to be increasing/decreasing
    Intermediate Value Theorem for derivatives
    L'Hospital's Rule
  4. Taylor's theorem