Existence of derivative at a point.

Question

Assume that f : (a, b) → R is differentiable on (a, b) except possibly at c ∈ (a, b). Assume that lim$_{x→c}$ f '(x) exists. Prove that f '(c) exists and f ' is continuous at c.

I am getting a counter example for this statement. Consider the function f(x) defined on (-1,1) by

f(x) = x , x ∈ (-1,0) U (0,1)

    5 , x = 0

Here f '(x) is 1 for all x ∈ (-1,0) U (0,1).Thus lim$_{x→0}$ f '(x) exists. But we know f(x) is not continuous at 0. Thus , f'(0) doesn't exist.

I don't know where I have made a wrong assumption in the example. Please help.

Answer 1

You're right. For the statement to be true, you need to add the assumption that $f$ is continuous at $c$ (and then you can prove it using the mean value theorem, as has been done many times on this site, for example here).