Short answers, pulled from the story.
Mathematical analysis is the branch of mathematics dealing with continuous functions, limits, and related theories such as differentiation, integration, measure, infinite sequences, series, and analytic functions. These theories are usually studied in the context of real and complex numbers and functions. Analysis evolved from calculus.
Mathematical analysis formally developed in the 17th century during the Scientific Revolution. The modern foundations were established in 17th century Europe, with Descartes's publication of La Géométrie in 1637 considered the establishment of the field. Newton and Leibniz independently developed infinitesimal calculus a few decades later.
In 1821, Cauchy began putting calculus on a firm logical foundation by rejecting the principle of the generality of algebra used by earlier mathematicians such as Euler. He introduced the Cauchy sequence and started the formal theory of complex analysis. Weierstrass and others later developed the epsilon-delta definition of limit.
The main branches of mathematical analysis include real analysis, complex analysis, functional analysis, harmonic analysis, differential equations, measure theory, numerical analysis, and vector analysis. Real analysis deals with real-valued functions of a real variable, while complex analysis investigates functions of complex numbers.
Ancient mathematicians used early ideas of analysis informally. Eudoxus and Archimedes used the method of exhaustion to compute area and volume, Liu Hui used the method of exhaustion in the 3rd century CE to find the area of a circle, and Madhava of Sangamagrama developed infinite series expansions in the 14th century.
Measure theory is the part of mathematical analysis that assigns a number representing size to suitable subsets of a set, generalizing length, area, and volume. The Lebesgue measure of the interval in the real numbers equals its everyday length, specifically 1. Lebesgue introduced Lebesgue integration, a major improvement over Riemann's theory of integration.