Real Analysis: Continuity, Uniform Continuity, and Convergence

Understanding the deep theoretical concepts of real analysis can feel overwhelming when transitioning from basic calculus to rigorous proofs. This written course demystifies the core pillars of real analysis—continuity, uniform continuity, and convergence—by breaking down abstract definitions into clear, logical steps. You will transition from memorizing formulas to confidently constructing mathematical proofs and applying systematic problem-solving strategies to complex analysis problems. What you'll learn: - Understand the foundational definitions of limits, continuity, and sequential convergence. - Distinguish between pointwise continuity and uniform continuity using clear criteria and counterexamples. - Apply convergence tests to sequences and series of functions with precision. - Master structured problem-solving frameworks to tackle rigorous mathematical proofs. - Analyze real-valued functions to determine their topological and analytical properties. The course begins with essential terminology and foundational definitions of real numbers and limits, before guiding you through detailed text-based explanations, analytical proofs, and step-by-step solved problems. This course is designed for undergraduate mathematics students, educators, and anyone preparing for exams in advanced mathematics. No advanced prior knowledge of real analysis is required, though a basic understanding of calculus is helpful. Start reading today to master the core concepts of real analysis and elevate your mathematical reasoning.