Real Analysis I

Continuity and limits:

• Definition and sequential characterization

• Property of continuous functions

• Uniform continuity

• Monotonicity and continuity

Differentiation:

• Definition, product, quotient and chain rules

• Rolle, mean value and Darboux theorems

• Taylor developments

Riemann-Stieltjes integral:

• Definition, integrable functions andproperties

• Fundamental theorem of calculus, integration by parts.

• Mean value theorem

Sequence and series of functions:

• Pointwise and uniform convergence

• Integration,  differentiation and uniform convergence

• Power series

• Stone-Weirstrass theorem

Exams:

• Practice midterm 1. Solutions. Practice Midterm 2. Solutions.

• Midterm 1. Solutions. Midterm 2. Solutions

• Practice final 1. Solutions. Practice final 2. Solutions

• Final 1. Solutions. Final 2. Solutions