For example, the derivative of continuous function is not necessarily continuous F (x) is differentiable everywhere on (a, b), but there exists x0 ∈ (a, b), so that f '(x0) exists, but f' (x) is discontinuous at x0 Who can give such an example?

For example, the derivative of continuous function is not necessarily continuous F (x) is differentiable everywhere on (a, b), but there exists x0 ∈ (a, b), so that f '(x0) exists, but f' (x) is discontinuous at x0 Who can give such an example?

Function f (x): when x is not equal to 0, f (x) = x ^ 2Sin (1 / x); when x = 0, f (x) = 0. This function is differentiable at (- ∞, + ∞). The derivative is f '(x): when x is not equal to 0, f' (x) = 2xsin (1 / x) - cos (1 / x); when x = 0, f '(x) = Lim {[f (x) - f (0)] / (x-0), X - > 0} = Lim [xsin (1 / x), X - > 0] = 0