f '(x)=αx^(α- 1)sin( 1/x^β)-βx^(α-β- 1)cos( 1/x^β)

When α- 1 >: 0, α-β-1>; 0,

Namely α > 1，0 & ltβ& lt； When α- 1, f'(x) = 0 and f'(0) = 0,

F'(x) is continuous when x = 0.