Editing Darboux integral (section)

{{Short description|Integral constructed using Darboux sums}}
{{Refimprove|date=February 2013}}
In [[real analysis]], the '''Darboux integral''' is constructed using '''Darboux sums''' and is one possible definition of the [[integral]] of a [[function (mathematics)|function]]. Darboux integrals are equivalent to [[Riemann integral]]s, meaning that a function is Darboux-integrable if and only if it is Riemann-integrable, and the values of the two integrals, if they exist, are equal.<ref>{{cite book|author1=David J. Foulis|author2=Mustafa A. Munem|title=After Calculus: Analysis|url=https://books.google.com/books?id=kSMnAQAAIAAJ|year=1989|publisher=Dellen Publishing Company|isbn=978-0-02-339130-9|page=396}}</ref> The definition of the Darboux integral has the advantage of being easier to apply in computations or proofs than that of the Riemann integral. Consequently, introductory textbooks on [[calculus]] and real analysis often develop Riemann integration using the Darboux integral, rather than the true Riemann integral.<ref>{{Cite book|title=Calculus (3rd. edition)|url=https://archive.org/details/calculus00spiv_191|url-access=limited|last=Spivak|first=M.|publisher=Publish Or Perish, Inc.|year=1994|isbn=0-914098-89-6|location=Houston, TX|pages=[https://archive.org/details/calculus00spiv_191/page/n266 253]–255}}</ref>  Moreover, the definition is readily extended to defining [[Riemann–Stieltjes integration]].<ref>{{Cite book|title=Principles of Mathematical Analysis (3rd. edition)|url=https://archive.org/details/principlesmathem00rudi_663|url-access=limited|last=Rudin|first=W.|publisher=McGraw-Hill|year=1976|isbn=007054235X|location=New York|pages=[https://archive.org/details/principlesmathem00rudi_663/page/n128 120]–122}}</ref>  Darboux integrals are named after their inventor, [[Gaston Darboux]] (1842–1917).