= Real analysis
{wiki}
Real analysis studies <limits>, <continuity>, <differentiation>, <integration>, and convergence of <sequences> and <functions> taking values in the <real numbers>. Its central question is how finite approximations determine an exact answer: what does an arbitrarily small error mean, and which operations respect that error? A <derivative> is a <limit> of quotients, while a <Riemann integral> is obtained from increasingly fine <Riemann sums>. Both need conditions that make the answer independent of the approximations chosen.
Two organizing ideas are <completeness of the real numbers>, which ensures that suitable approximations have a destination, and <uniform convergence>, which controls approximation error over an entire domain. <Pointwise convergence> supplies separate guarantees at separate inputs; <uniform convergence> supplies one guarantee for all inputs. This distinction explains why replacing an approximating <function> by its <limit> can preserve <continuity> or <integrals> under appropriate hypotheses, yet fail without them.
For historical context, Cauchy's 1821 \i[Cours d'analyse] explicitly discusses convergence of <series>. Its Note VII treats double <series> and compares summing by rows and by columns under stated convergence assumptions: controlling an infinite process was already a concrete problem. This is historical context for questions about <limits>, rather than an attribution of the modern definitions below. https://www.e-rara.ch/download/pdf/8543076.pdf[Cauchy, Cours d'analyse (1821), Note VII, pp. 537–539, original scan]{ref}.
Back to article page
