Analysis grew from calculus into the study of convergence itself
Mathematical analysis studies functions, spaces, and operators through approximation and convergence. It grew out of calculus’s derivatives and integrals, formally blooming in seventeenth-century Europe’s Scientific Revolution while older cultures already summed series. Today it spans real and complex variables, measure theory and spaces of functions.
Mathematical analysis asks how functions, sequences and operators behave under limits and small perturbations: whether they converge and stay continuous, how regular and stable they are, and how accurate an approximation is. The standard setting is the real numbers, whose completeness, often stated as the least-upper-bound property, underpins results on limits, continuity, differentiation and integration. Over time the same methods reached functionals and operators.
Early hints appear in the Jain Kalpasūtra of Bhadrabāhu, dated to 433 BCE, which uses the sum of a geometric series. Medieval Islamic mathematicians pushed further, Ibn al-Haytham tackling area problems and power sums and Ibrahim ibn Sinan broadening the techniques of Archimedes, while European philosophers argued, following Aristotle, over whether the continuum could be made of points. In the early 17th century Kepler used infinitesimal sums for areas and volumes, Galileo tied mathematics to motion, Cavalieri devised the method of indivisibles and Torricelli extended it.
Newton and Leibniz, working independently late in that century, merged tangent problems, quadrature and their inverses into a single calculus. In the 18th century Euler put the function at the centre, and Lagrange and others developed power series, differential equations and the calculus of variations. Bolzano's 1816 definition of continuity marked the birth of real analysis, though it stayed obscure until the 1870s, and from 1821 Cauchy began placing calculus on firm logical ground.
Late in the 19th century Dedekind built the real numbers from cuts that fill the gaps between rationals, while so-called monsters such as continuous but nowhere differentiable functions and space-filling curves drew attention. Jordan developed a theory of measure, Cantor founded naive set theory and Baire proved his category theorem. Convergence alone says nothing about how good an approximation is, so results such as Taylor's theorem bound the error of a linear approximation using the second derivative.
Source: Mathematical analysis
