## Department of Mathematics Syllabus

This syllabus is advisory only. For details on a particular instructor's syllabus (including books), consult the instructor's course page. For a list of what courses are being taught each quarter, refer to the Courses page.

MAT 127B: Real Analysis

**Approved:**2015-11-06, John Hunter and Janko Gravner

**Suggested Textbook:**(actual textbook varies by instructor; check your instructor)

Stephen Abbott, Understanding Analysis, 2nd ed., Springer 2015. (Alternative: Kenneth Ross, Elementary Analysis, Springer, 2nd ed., Springer, 2013.)

https://www.math.ucdavis.edu/~hunter/intro_analysis_pdf/intro_analysis.html

https://www.math.ucdavis.edu/~hunter/intro_analysis_pdf/intro_analysis.html

**Course Description:**

Derivatives, integrals, sequences of functions, and power series.

**Suggested Schedule:**

Suggested number of lectures are indicated in parentheses.

- (2) Derivatives. Algebraic properties.
- (1) Chain rule.
- (1) Extreme value theorem.
- (2) Mean value theorem. Consequences.
- (1) Sequences and series of functions.
- (3) Pointwise and uniform convergence. Continuity of uniform limits.
- (1) Power series. Radius of convergence.
- (2) Definition of the Riemann integral.
- (3) Properties of the Riemann integral.
- (2) Fundamental theorem of calculus.
- (1) Exchange of integral and limits.
- (2) Taylor series. Integral form and Lagrange of the remainder.
- (2) Integration and differentiation of power series.
- (2) Improper integrals.
- (1) Integral test for series.

Total: 26 lectures