The correct option is
A f is differentiable at
x=0.f(x)=x3sgn(x)⇒f(x)=⎧⎪⎨⎪⎩x3,x>00,x=0−x3,x<0
Only point at which f(x) may be discontinuous is x=0
limx→0−f(x)=limx→0+f(x)=f(0)=0
So, f(x) is continuous at x=0
Now,
f′(x)={3x2,x>0−3x2,x<0⇒f′(0−)=f(0+)=0
So, f(x) is differentiable at x=0