Analysis is the part of mathematics that primarily deals with continuity, as opposed to discreteness. While there is no way to define exactly what is "analysis" and what is not, one can be pretty sure that something has an analytical flavor every time he hears such words as limit, integral, derivative, series, 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 and ending with the Lebesgue theory of measure and integration and functional analysis.
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