IIT JAM Follow
October 19, 2020 11:20 am 30 pts
Ex: Let f be the function defined on R by setting fx) [x]. for all x e R, where [x] denotes the greatest integer not exceeding x. Show that f is discontinuous at the noi X =0, t 1, + 2, t 3,. and is continuous at every other point. Hence f is continuous at s the points Sol. By definition, we have x = 0, for 0 x< 1, x] = 1, for 1x < 2, x) = 2, for 2 sx < 3, x]=-1, for -1 x <0. X=-2 for -2 x - 1 and so on. At x 0 lim f(x) = lim [x] =-1 X0 (x <0) X0- lim f(x) lim [x]=0. X0 (x>0) X0+ Since lim f(x) # lim fx), f is discontinuous at 0. X0 X0+ At x = 1 lim f(x) = lim [x]= -2, X1- X 1 (x<1) Jlim f(x)= lim [x]=1, X1+ X-1 (x<-1) So f is discontinuous at -1. Similarly, f is discontinuous at - 2, - 3, -4, Let a e R Z be any real number but not an integer. Then there exists an integer n such that