Analysis is the part of mathematics that primarily deals with definitions and theorems concerning continuity and convergence. Analysis covers a wide variety of topics which appear quite different to each other, so there is no way to define exactly what is "analysis" and what is not. However, one can be pretty sure that something has an analytical flavor every time he or she hears such words as limit, integral, derivative, series, function etc.
The foundations of mathematical analysis as we know it today were laid in 17-20th centuries starting with the development of integral and differential calculus by Newton and Leibnitz. However, earlier instances of the methods of analysis were seen in Zeno's paradox of the dichotomy and Eudoxus and Archimedes method of exhaustion.
The current stage of mathematical analysis was set by Bolzano when he introduced the modern epsilon delta definition of continuity and Cauchy who tried to put calculus on a "firm logical basis."
Currently analysis has grown into a wide and varied area of study. So large is the "umbrella" of analysis that two researchers working under it might have a hard time conversing about their respective research. That being said there are some general, (but hardly exhaustive), subdivisions of analysis that most mathematicians accept.
Real Analysis- This includes what one would learn in a beggining analysis class; ideas such as continuity, connectedness, compactness, convergence,etc. Also included under this umbrella are modern integration theories (such as those by lebesgue and darboux). Such theories form the axiomatic foundation for modern probability.
Functional Analysis- Studies spaces of functions. More generally, functional analysis studies what are essentially infinite dimensional vector spaces and its generalizations.
Harmonic analysis- Deals with fourier analysis, but also operators (generalizations of matrices) on the underlying spaces in which much of the fourier analysis takes place.
Complex Analysis- Studying the generalizations of real analysis to the complex numbers.
There are some other branches of analysis but are more applications and would be better discussed in their respective topics. These topics include p-adic analysis, nonlinear analysis, non-standard analysis, microlocal analysis, and numerical analysis.
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