Question

If f(x) = {(x2sin(1/x), x ≠ 0), (0, x = 0) then

(A) f and f' are continuous at x = 0

(B) f is derivable at x = 0

(C) f and f' are derivable at x = 0

(D) f is derivable at x = 0 and f' is continuous at x = 0

Answer 1

Answer is (B) f is derivable at x = 0

When x ≠ 0

But lim(x→0) cos(1/x) does not exist, so lim(x→0) f'(x) does not exist.

Hence f' is not continuous (so not derivable) at x = O.