Note that if there is a sequence (xn)n2Nwithxn---!n!1afor which (f(xn))n2Ndoesn’tconverge at all, thenfis discontinuous ata. One way to exploit this is to find a sequencexn---!n!1afor whichf(xn) has two di↵erent subsequential limits. For example, iff(x) = sgn (x) =8><>:-1,ifx <0;0,ifx= 0;1,ifx >0,we takexn= (-1)n1n---!n!10, thenf(xn) = (-1)n, which is divergent since it has twosubsequential limit points. Thus, the signum function is discontinuous atx= 0.Using the properties of the limit from Theorem 5.10 we immediately derive the usualproperties of continuous functions from calculus:Theorem 5.18.Assumef, g:A✓R!Rare each continuous ata2A. Then(f+g),fg, andf/g(providedg(a)6= 0) are all continuous ata2A.The other natural combination between continuous functions iscomposition. To definethe composition of two functionsh(x) =g◦f(x) =g(f(x)) we have to be sure that theirdomain and range are compatible.Theorem 5.19.Letf:A✓R!Randg:B✓R!Rwithf(A)✓B.Iffiscontinuous ata2Aandgis continuous atb=f(a)2B, thenh=g◦fis continuous ata.Proof.Using the definition of continuity forgaty=b, for any">0 there existsγ>0 forwhich|g(y)-g(b)|<"for ally2Bwith|y-b|<γ.