Continuity of f(x):f(0+)=h2⋅sinh1=0f(0−)=(−h)2⋅sin(h−1)=0f(0)=0f(x) is continuousf′(0+)=h→0limhf(0+h)−f(0)=hh2⋅sin(h1)−0=0f′(0−)=h→0lim−hf(0−h)−f(0)=−hh2⋅sin(−h1)−0=0f(x) is differentiable.f′(x)=2x⋅sin(x1)+x2⋅cos(x1)⋅x2−1f′(x)={2x⋅sin(x1)−cos(x1),0,x=0x=0 (x) is not continuous (as cos(x1) is highly oscillating at x=0 )