If the derivative exists when $x$ approaches a point, is $f$ differentiable at that point?

Question

I'm having a hard time trying to figure if a given statement is true or not:

If $f$ is defined in a neighborhood of $x_0$ and $\lim_{x\to x_0}f'(x)$ exists, then $f$ is differentiable at $x_0$.

Does this always hold, or is there a counter example?

Thanks.

Answer 1

No, consider $f(x) = x$ for all $x\neq 0$ and $f(0)=1$. Then, the $f(x)$ is defined for all $x$, $\lim\limits_{x\rightarrow 0} f'(x) = 1$, but $f$ is not continuous at $x=0$ and thus cannot be differentiable.

Answer 2

There are several examples in two-dimensional case, too. The existence of partial derivatives of a function on a point does not imply that it is differentiable on that point.

Answer 3

The answer is no. You may have noticed that in the simple counterexamples that have been given, $f$ cannot be differentiable at $x_0$ because $f$ is not even continuous at $x_0$. I'm surprised that nobody has pointed out that there's a reason for that:

The answer is yes if you also assume that $f$ is continuous at $x_0$.

Hint: Apply the Mean Value Theorem to $$\frac{f(x_0+h)-f(x_0)}{h}.$$