Let f(x)=x|x| . The set of points where f(x) is twice differentiable is :

Since f(x)=x|x|∴f(x)==−x2,ifx<0=0,ifx=0=x2,ifx>0The function f  is differentiable everywhere and its derivative is given by f'(x)=−2xifx<00ifx=02xifx>0f"(0+)=limh→0f'(0+h)−f'(0)h=limh→02h−0h=2f"(0−)=limh→0−f(0−h)−f'(0)−h=limh→0−2(−h)−0−h=−2 Hence f is not twice diff. at x=0 But   f''(x)=-2, if   x<0=2, if   x>0 Thus, the set of points. Where f is twice differentiable is (-∞,-∞)-{0} .