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In [[probability theory]], the '''continuous mapping theorem''' states that continuous functions [[Continuous function#Heine definition of continuity|preserve limits]] even if their arguments are sequences of random variables. A continuous function, in [[Continuous function#Heine definition of continuity|Heine’s definition]], is such a function that maps convergent sequences into convergent sequences: if ''x<sub>n</sub>'' → ''x'' then ''g''(''x<sub>n</sub>'') → ''g''(''x''). The ''continuous mapping theorem'' states that this will also be true if we replace the deterministic sequence {''x<sub>n</sub>''} with a sequence of random variables {''X<sub>n</sub>''}, and replace the standard notion of convergence of real numbers “→” with one of the types of [[convergence of random variables]].
This theorem was first proved by [[Henry Mann]] and [[Abraham Wald]] in 1943,<ref>{{cite journal | doi = 10.1214/aoms/1177731415 | last1 = Mann |first1=H. B. | last2=Wald |first2=A. | year = 1943 | title = On Stochastic Limit and Order Relationships | journal = [[Annals of Mathematical Statistics]] | volume = 14 | issue = 3 | pages = 217–226 | jstor = 2235800
==Statement==
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