Analysis I

This course focuses on the structure of the real line and functions on the reals. Notions of convergence and continuity are studied and also some important results in calculus about these concepts, and the properties of the real line that lead to these results.

This is the first of the three-courses in analysis. The course is a study of properties of the system of real numbers, R, and of continuous real-valued functions of one real variable. The course intends to develop geometric intuition, skills of analytic estimation, numerical computations, and of translating intuition into rigorous arguments. With an axiomatic introduction to the system of real numbers, the first part of the course deals with its algebraic properties, order properties and the consequences of the completeness property. Several facts related to real numbers, that are assumed at school level, are established using these properties. The geometric ideas underlying these properties are highlighted, and visualisation as a skill is nurtured throughout the course.

The course then deals with sequences of real numbers, introduced in Explorations in Mathematics 2, in a rigorous manner while the notion of series is only introduced as a special case of sequence. The last part concerns the notion of continuity of functions from R to R, fundamental properties such as the Intermediate Value Theorem, boundedness of continuous functions defined on closed and bounded intervals, etc. and their consequences.

To ensure smooth transitions from IBL to more traditional pedagogy, a few topics at the start of the course will be dealt partially through active learning strategies and partially through traditional lectures-cum-discussions approach.