Let  be such that the functionfx=cos−11−x2sin−11−xx−x3,x≠0 α,                                                              x=0          is Continuous at  x=0,where x=x−x,x  is the greatest integer less than or equal to x. Then :

RHL=limx→0+cos-11-x2sin-1(1-x)x1-x2=π2limx→0+cos-11-x2x=π2limx→0+-11-1-x22(-2x) (L'Hospital Rule) =πlimx→0+x2x2-x4=πlimx→0+12-x2=π2LHL=limx→0-cos-11-(1+x)2sin-1(-x)(1+x)-(1+x)3=π2limx→0-sin-1x(1+x)(1+x)2-1=π2limx→0+sin-1xx2+2x=π212=π4 As LHL≠RHL so f(x) is not continuous at x=0.There is no such α such exists