Real analysis

Real analysis is a field in mathematics that focuses on the set of real numbers, their properties, sequences and functions. Included in this branch of mathematics is concepts of limits and convergence, calculus, and properties of real-valued functions such as continuity.

The foundations of real analysis were shaken at the end of the 19th century with the work of Richard Dedekind, who relaid the Archimedean-style groundwork with his own radical concepts. His Dedekind cuts undercut the assumption of the continuity of the real line, by cutting at gaps between points. Mathematicians were worried that his techniques used the dubious Axiom of Choice and seemingly non-elementary methods. However, the unifying ideas of Cauchy, specifically that of the Cauchy sequence and completeness, helped eliminate doubts and gain acceptance for Dedekind's ideas among real analysts. Dedekind cuts are now viewed as a solid foundation for real analysis, more than Archimedes' ideas ever were.