Real analysis

The least-upper-bound property (completeness) of the real numbers: that every non-empty bounded above subset has a supremum. Virtually every major theorem in real analysis relies on the completeness of the real numbers, including those concerning the existence of limits of sequences, the behaviour of continuous functions, and the existence and uniqueness of solutions to differential equations.

Real analysis is the part of mathematical analysis, especially as taught in undergraduate and graduate courses, that develops calculus rigorously over the real numbers and Euclidean spaces. Introductory real analysis is sometimes called advanced calculus, and studies limits, continuity, compactness, differentiation, integration, and series. More advanced courses often include measure theory, Lebesgue integration, and function spaces. Real analysis is also known, especially in older books, as the theory of functions of a real variable, in contrast to the theory of complex variables.

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