Proof of Equivalence of Continuity of Mapping of an Element of an Open Set to an Element of an Open Set
Thenis continuous if and only ifthere exists an open setwith
The converse is also true.
Supposeis continuous. Take anyand any open neighbourhoodof
Sinceis continuousSincewe have
Letrepresent an open set ofSupposethensois an open neighbourhood ofAn open setexists such thatand
Buthence for eachwhere
Henceis the union of open subsets ofand so is an open subset.