Shahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-58538120180101On the pointfree counterpart of the local definition of classical continuous maps1832712ENBernhard BanaschewskiDepartment of Mathematics and Statistics, McMaster University, Hamilton, ON L8S 4K1, Canada.Journal Article20160707The familiar classical result that a continuous map from a space $X$ to a space $Y$ can be defined by giving continuous maps $varphi_U: U to Y$ on each member $U$ of an open cover ${mathfrak C}$ of $X$ such that $varphi_Umid U cap V = varphi_V mid U cap V$ for all $U,V in {mathfrak C}$ was recently shown to have an exact analogue in pointfree topology, and the same was done for the familiar classical counterpart concerning finite closed covers of a space $X$ (Picado and Pultr [4]). This note presents alternative proofs of these pointfree results which differ from those of [4] by treating the issue in terms of frame homomorphisms while the latter deals with the dual situation concerning localic maps. A notable advantage of the present approach is that it also provides proofs of the analogous results for some significant variants of frames which are not covered by the localic arguments.http://cgasa.sbu.ac.ir/article_32712_7102051b8b0d2b0555b4ab6cee021fc7.pdf