Question

Exercise 4.3.3. (a) Supply a proof for Theorem 4.3.9 using the ed charac- terization of continuity. Excreien: :03a supely a pot be Tovem 130 ming the ó dheas (b) Give another proof of this theorem using the sequential characterization of continuity (from Theorem 4.3.2 (iii)).

only (b) please

Answer 1

Liet f. A R and 9: B R and ut range off is contained in the domain B of g I get f is continuous atc and g is continuous nous at f(c) EB To show ! Gof is continuous at c using sequential I criterion .. so let Xnc Then to snool go flen) = (g(fean)) at xal nem gf (x) = gfec), as f is continuous ie g fexani - gfors 3, faxno=tke) is continous continuous at e.