mathematical analysis
Calculus tells you how to compute a derivative or an integral; analysis tells you why those operations are legitimate and exactly when they work. Think of calculus as a powerful set of recipes and analysis as the kitchen science that explains why the recipes do not blow up — and warns you about the rare ingredients that would.
Mathematical analysis is the branch of mathematics that studies limits, continuity, differentiation, integration, sequences, series and the structure of the spaces on which these live, all developed by careful deductive proof from a small set of axioms about the real numbers. Its central engine is the limit, made precise through the epsilon–delta language, and almost every theorem is a guarantee of the form ‘under these hypotheses, this limiting process behaves as expected’.
The subject is honest about its own boundaries: it does not merely assert that integrals and derivatives exist, it characterises precisely when they do and exhibits counterexamples when a tempting statement fails. From this core (real analysis) grow complex analysis, measure theory, functional analysis and beyond — the same rigorous spirit applied to ever richer objects.
Calculus says the derivative of x^2 is 2x. Analysis asks: what does the limit (f(x+h) - f(x))/h as h -> 0 actually mean, and proves that for f(x) = x^2 it equals 2x for every real x by an epsilon–delta argument, not by hand-waving about ‘h becoming zero’.
Same answer as calculus, but earned by proof rather than assumed.
Analysis is not ‘calculus with more Greek letters’. The shift is conceptual: from manipulating formulas to proving the manipulations are valid, with the limit as the unifying idea.