Real Analysis: Foundations

Part I – Sequences, Series, Limits, and Continuity
• Numerical sequences: convergence, divergence, limits of sequences, and basic limit theorems.
• Numerical series: convergence, divergence, and convergence tests.
• Real-valued functions: limits of functions, continuity, extreme value property, and intermediate
value property.

Part II – Differentiation and Multivariable Calculus
• Derivatives and differentials, rules of differentiation, and fundamental theorems of differential
calculus.
• Higher-order derivatives and Taylor expansions.
• Qualitative study of functions: monotonicity, local and global extrema, concavity and
convexity, asymptotic behaviour, graph sketching.
• Functions of several variables: limits in Rn, continuity, partial derivatives, and differentiability.

Part III – Integration
• The Riemann integral: definition and existence, integrable functions, and properties of the
integral.
• Fundamental Theorem of Calculus.
• Techniques of integration: substitution, integration by parts, and integration of rational
functions using partial fractions.
• Improper integrals.
• Comprehensive review exercises.