Editing Continuous function (section)

==History==

A form of the [[Limit_of_a_function#(ε,_δ)-definition_of_limit|epsilon–delta definition of continuity]] was first given by [[Bernard Bolzano]] in 1817. [[Augustin-Louis Cauchy]] defined continuity of <math>y = f(x)</math> as follows: an infinitely small increment <math>\alpha</math> of the independent variable ''x'' always produces an infinitely small change <math>f(x+\alpha)-f(x)</math> of the dependent variable ''y'' (see e.g. ''[[Cours d'Analyse]]'', p.&nbsp;34). Cauchy defined infinitely small quantities in terms of variable quantities, and his definition of continuity closely parallels the infinitesimal definition used today (see [[microcontinuity]]). The formal definition and the distinction between pointwise continuity and [[uniform continuity]] were first given by Bolzano in the 1830s, but the work wasn't published until the 1930s. Like Bolzano,<ref>{{cite web|url=http://dml.cz/handle/10338.dmlcz/400352|title=Rein analytischer Beweis des Lehrsatzes daß zwischen je zwey Werthen, die ein entgegengesetzetes Resultat gewähren, wenigstens eine reelle Wurzel der Gleichung liege |year=1817 |last1=Bolzano |first1=Bernard |publisher=Haase|location=Prague}}</ref> [[Karl Weierstrass]]<ref>{{Citation | last1=Dugac | first1=Pierre | title=Eléments d'Analyse de Karl Weierstrass | journal=Archive for History of Exact Sciences | year=1973 | volume=10 | issue=1–2 | pages=41–176 | doi=10.1007/bf00343406| s2cid=122843140 }}</ref> denied continuity of a function at a point ''c'' unless it was defined at and on both sides of ''c'', but [[Édouard Goursat]]<ref>{{Citation | last1=Goursat | first1=E. | title=A course in mathematical analysis | publisher=Ginn | location=Boston | year=1904 | page=2}}</ref> allowed the function to be defined only at and on one side of ''c'', and [[Camille Jordan]]<ref>{{Citation | last1=Jordan | first1=M.C. | title=Cours d'analyse de l'École polytechnique | publisher=Gauthier-Villars | location=Paris | edition=2nd |year=1893 | volume=1|page=46|url={{Google books|h2VKAAAAMAAJ|page=46|plainurl=yes}}}}</ref> allowed it even if the function was defined only at ''c''. All three of those nonequivalent definitions of pointwise continuity are still in use.<ref>{{Citation|last1=Harper|first1=J.F.|title=Defining continuity of real functions of real variables|journal=BSHM Bulletin: Journal of the British Society for the History of Mathematics|year=2016|volume=31|issue=3|doi=10.1080/17498430.2015.1116053|pages=1–16|s2cid=123997123}}</ref> [[Eduard Heine]] provided the first published definition of uniform continuity in 1872, but based these ideas on lectures given by [[Peter Gustav Lejeune Dirichlet]] in 1854.<ref>{{citation|last1=Rusnock|first1=P.|last2=Kerr-Lawson|first2=A.|title=Bolzano and uniform continuity|journal=Historia Mathematica|volume=32|year=2005|pages=303–311|issue=3|doi=10.1016/j.hm.2004.11.003|doi-access=}}</ref>