real analysis

Real analysis is the study of the real number line and everything that lives on it — sequences, functions, derivatives, integrals — done rigorously. If analysis is a city, real analysis is its foundation and old town: the streets everything else was built outward from.

Concretely, real analysis develops the theory of limits, continuity, differentiation and Riemann (and later Lebesgue) integration over the real numbers R, starting from the axioms that make R a complete ordered field. The single property that distinguishes R from the rationals Q — completeness, the absence of gaps — is what makes the central theorems (intermediate value, extreme value, convergence of Cauchy sequences) true.

Real analysis is foundational in a precise sense: complex analysis, measure theory, functional analysis and differential equations all rest on its results and inherit its standards of proof. The honest point worth stressing: nearly every difficulty and counterexample students meet later already appears, in miniature, in real analysis on the line — which is exactly why it is taught first.