Question

prove that continuity is necessary but not a sufficient condition for the existence of a finite derivatives.

Answer 1

Answer:

Same for the other side. So if the limits of the derivatives exist and are equal then f is differentiable. The other implication is false because f′ could exist everywhere but not be continuous at c. The function f(x)=x2sin1/x if x≠0 and f(0)=0 is differentiable everywhere but the derivative is discontinuous. Even worse you could take f(x)=x2 if x is rational and f(x)=0 otherwise. then f is differentiable at x=0 and discontinuous at every other point.