Calculus or Analysis II Notes

These notes provide a rigorous development of topology, functions, and the calculus of a single variable. Beginning with the topological structure of adherence, open sets, and the classification of discontinuities, we establish the foundations of continuity through the epsilon-delta formalism. We explore the global consequences of compactness and connectedness, establishing the Heine-Borel theorem and the topological origins of the Intermediate Value Theorem. The development proceeds through a linear-approximation treatment of differentiation and an axiomatic approach to the Riemann integral, culminating in the Riemann-Lebesgue characterization of integrability.

Notes

Real Analysis II

Note: While these notes provide a self-contained path through analysis-based calculus, I highly recommend consulting Stephen Abbott’s Understanding Analysis (Chapters 3–7) and Michael Spivak’s Calculus (Part III) to gain a comprehensive perspective. Specifically, Spivak’s text is invaluable for its mechanically intense and challenging problem sets; I suggest working through the exercises in my notes first to establish a theoretical foundation before tackling Spivak’s questions (cause we all need the tears).

Prerequisites

(Recommended for induction, field axioms, sequences)

• Algebra Notes
• Logic Notes
• Set Theory
• Real Analysis I

Recommendations

• Understanding Analysis - Stephen Abbott (Highly recommended for intuition)

• Calculus by Spivak (Highly recommended for just the problem sets)