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'''Real analysis''' is a field in [[mathematics]] that focuses on the set of [[real number]]s, their properties, [[sequence]]s and [[function]]s.  Included in this branch of mathematics is concepts of [[limit]]s and [[convergence]], [[calculus]], and properties of real-vaalued functions such as [[continuous|continuity]].

'''Real analysis''' is a field in [[mathematics]] that focuses on the set of [[real number]]s, their properties, [[sequence]]s and [[function]]s.  Included in this branch of mathematics is concepts of [[limit]]s and [[convergence]], [[calculus]], and properties of real-vaalued functions such as [[continuous|continuity]].

The first serious consideration of the real numbers was by [[Archimedes]] and followed by other [[Greek]]s such as [[Euclid]], [[Pappus]], and [[Zeno]]. To honor Archimedes' contribution, real analysts have named a property of the real numbers the [[Archimedean property]]. Real analysis remained in [[geometry]]'s shadow until the development of the subfield of [[calculus]]. This subject [[coordinatization|coordinatized]] all known geometry, subsuming it into its scope.

The first serious consideration of the real numbers was by [[Archimedes]] and followed by other [[Greek]]s such as [[Euclid]], [[Pappus]], and [[Zeno]]. To honor Archimedes' contribution, real analysts have named a property of the real numbers the [[Archimedean|Archimedean property]]. Real analysis remained in [[geometry]]'s shadow until the development of the subfield of [[calculus]]. This subject [[coordinatization|coordinatized]] all geometry known at the time, subsuming it into its scope.

The foundations of real analysis were shaken at the end of the 19th century with the work of [[Dedekind]]. His [[Dedekind_cut]]s questioned the continuity of the real line, by ''cutting'' at ''gaps'' between points. However, when it became apparent that his techniques used the dubious [[Axiom of Choice]], his concerns were dismissed by real analysts as [[elementary_proof|non-elementary]].

The foundations of real analysis were shaken at the end of the 19th century with the work of [[Dedekind]]. His [[Dedekind cut]]s questioned the continuity of the real line, by ''cutting'' at ''gaps'' between points. However, when it became apparent that his techniques used the dubious [[Axiom of Choice]], his concerns were dismissed by real analysts as [[elementary_proof|non-elementary]].

[[category:mathematics]]

[[category:mathematics]]