Synopsis for Analysis II: Continuity and Differentiability

Overview

In this term's lectures, we study continuity of functions of a real or complex variable, and differentiability of functions of a real variable.

Learning Outcomes

At the end of the course students will be able to apply limiting properties to describe and prove continuity and differentiability conditions for real and complex functions. They will be able to prove important theorems, such as the Intermediate Value Theorem, Rolle's Theorem and Mean Value Theorem, and will continue the study of power series and their convergence.

Synopsis

Definition of the function limit. Examples and counter examples to illustrate when (and when it doesn't). Definition of continuity of functions on subsets of and in terms of and . The algebra of continuous functions; examples, including polynomials. Continuous functions on closed bounded intervals: boundedness, maxima and minima, uniform continuity. Intermediate Value Theorem. Inverse Function Theorem for continuous strictly monotone functions.

Sequences and series of functions. Uniform limit of a sequence of continuous functions is continuous. Weierstrass's M-test for uniformly convergent series of functions. Continuity of functions defined by power series.

Definition of the derivative of a function of a real variable. Algebra of derivatives, examples to include polynomials and inverse functions. The derivative of a function defined by a power series is given by the derived series (proof not examinable). Vanishing of the derivative at a local maximum or minimum. Rolle's Theorem. Mean Value Theorem with simple applications: constant and monotone functions. Cauchy's (Generalized) Mean Value Theorem and L'Hôpital's Formula. Taylor's Theorem with remainder in Lagrange's form; examples of Taylor's Theorem to include the binomial expansion with arbitrary index.

Reading

1. Robert G. Bartle, Donald R. Sherbert, Introduction to Real Analysis (Wiley, Third Edition, 2000), Chapters 4–8.
2. R. P. Burn, Numbers and Functions, Steps into Analysis (Cambridge University Press, 2000). [This is a book of problems and answers, a DIY course in analysis]. Chapters 6–9, 12.
3. Walter Rudin, Principles of Mathematical Analysis (McGraw-Hill, 3rd edition, 1976). Chapters 4,5,7.
4. J. M. Howie, Real Analysis, Springer Undergraduate Texts in Mathematics Series (Springer, 2001), ISBN 1-85233-314-6.

Alternative Reading

1. Mary Hart, A Guide to Analysis (MacMillan, 1990), Chapters 4,5.
2. J. C. Burkill, A First Course in Mathematical Analysis (Cambridge University Press, 1962), Chapters 3, 4, and 6.
3. K. G. Binmore, Mathematical Analysis A Straightforward Approach, (Cambridge University Press, second edition, 1990), Chapters 7–12, 14–16.
4. Victor Bryant, Yet Another Introduction to Analysis (Cambridge University Press, 1990), Chapters 3 and 4.
5. M. Spivak, Calculus (Publish or Perish, 3rd Edition, 1994), Part III.
6. Brian S. Thomson, Judith B. Bruckner, Andrew M. Bruckner, Elementary Analysis (Prentice Hall, 2001), Chapters 5–10.