Functions of One Real Variable for Competitive Mathematics

Mastering real analysis and the behavior of single-variable functions is a crucial milestone for anyone preparing for competitive mathematics and university-level entrance exams. This text-based course guides you from foundational set theory and real number properties to advanced calculus concepts. You will build a rigorous mathematical intuition, learning how to analyze functions, evaluate limits, and apply core theorems with precision. What you'll learn: 1. Understand the fundamental properties of real numbers, sequences, and series. 2. Analyze limits, continuity, and uniform continuity of single-variable functions. 3. Apply key differentiability theorems, including Mean Value Theorems and Taylor's theorem. 4. Master Riemann integration and the Fundamental Theorem of Calculus. 5. Solve complex mathematical problems systematically using proven analytical techniques. 6. Explore modern applications of single-variable optimization in computational algorithms. The course begins with essential definitions of real numbers and topology, gradually advancing through limits, continuity, differentiation, and integration, complete with step-by-step solved mathematical proofs. This course is designed for undergraduate students, mathematics enthusiasts, and aspirants preparing for competitive exams like IIT JAM. No advanced prerequisites are required, though a basic familiarity with algebra is helpful. Start reading today to build a flawless foundation in real analysis and ace your upcoming exams.