Let f(x)=[x] and g(x)=0,x∈Zx2,x∈R−Z ([.] represents the greatest integer function.) Then

Since limx→1− g(x)=limx→1+ g(x)=1 and g(1)=0g(x) is not continuous at x = 1 but limx→1 g(x) exists.we have limx→1− f(x)=limh→0 f(1−h)=limh→0 [1−h]=0and limx→1+ f(x)=limh→0 f(1+h)=limh→0 [1+h]=1So, limx→1 f(x) does not exist and, hence, f(x) is not continuous at x = 1.We have gof(x)=g(f(x))=g([x])=0∀x∈RSo, gof is continuous for all x.We have fog(x)=f(g(x))=f(0),x∈Zfx2,x∈R−Z=0,x∈Zx2,x∈R−Zwhich is clearly not continuous.