Editing Elementary function (section)

{{short description|A kind of mathematical function}}
{{About| |the complexity class | Elementary recursive function}}

In [[mathematics]], an '''elementary function''' is a [[function (mathematics)|function]] of a single [[variable (mathematics)|variable]] (typically [[Function of a real variable|real]] or [[Complex analysis#Complex functions|complex]]) that is defined as taking [[addition|sums]], [[multiplication|products]], [[algebraic function|roots]] and [[composition of functions|compositions]] of [[finite set|finitely]] many [[Polynomial#Polynomial functions|polynomial]], [[Rational function|rational]], [[Trigonometric functions|trigonometric]], [[Hyperbolic functions|hyperbolic]], and [[Exponential function|exponential]] functions, and their [[Inverse function|inverses]] (e.g., [[Inverse trigonometric functions|arcsin]], [[Natural logarithm|log]], or ''x''<sup>1/''n''</sup>).<ref name=":1">{{Cite book|title=Calculus|last=Spivak, Michael.|date=1994|publisher=Publish or Perish|isbn=0914098896|edition=3rd|location=Houston, Tex.|pages=363|oclc=31441929}}</ref>

All elementary functions are continuous on their [[Domain of a function|domains]].

Elementary functions were introduced by [[Joseph Liouville]] in a series of papers from 1833 to 1841.<ref>{{harvnb|Liouville|1833a}}.</ref><ref>{{harvnb|Liouville|1833b}}.</ref><ref>{{harvnb|Liouville|1833c}}.</ref>  An [[abstract algebra|algebraic]] treatment of elementary functions was started by [[Joseph Fels Ritt]] in the 1930s.<ref>{{harvnb|Ritt|1950}}.</ref> Many textbooks and dictionaries do not give a precise definition of the elementary functions, and mathematicians differ on it.<ref name=":0">{{Cite journal |last1=Subbotin |first1=Igor Ya. |last2=Bilotskii |first2=N. N. |date=March 2008 |title=Algorithms and Fundamental Concepts of Calculus |url=https://assets.nu.edu/assets/resources/pageResources/Journal_of_Research_March081.pdf |journal=Journal of Research in Innovative Teaching |volume=1 |issue=1 |pages=82–94}}</ref>