Ashmit Rao

Step functions, their integral, basic properties. Minorants and majorants of bounded functions on bounded intervals. Definition of Riemann integral. Elementary properties of Riemann integrals: positivity, linearity, subdivision of the interval.

The application of uniform continuity to show that continuous functions are Riemann integrable on closed bounded intervals; bounded continuous functions are Riemann integrable on bounded intervals.

The Mean Value Theorem for Integrals. The Fundamental theorem of Calculus; integration by parts and by substitution.

The interchange of integral and limit for a uniform limit of integrable functions on a bounded interval. Term-by-term integration and differentiation of a (real) power series (interchanging limit and derivative for a series of functions where the derivatives converge uniformly).