Let f is differentiable on R. Following statement is true?
If $\lim_{x\to a}f'(x)=\alpha$, then $f'(a)=\alpha$ (where $\alpha$ and $a $ are real numbers)
Asked
Active
Viewed 54 times
1
-
False. Take f to be the function that is 0 everywhere except at x=1 where it is 1. – cauchyproblem May 17 '16 at 00:00
-
@cauchyproblem That function isn't differentiable. – Git Gud May 17 '16 at 00:01
-
Ah ok, so you are assuming the function is differentiable at a. – cauchyproblem May 17 '16 at 00:04
2 Answers
0
By mean value theorem, there is $|c_x-a|<|x-a|$ s.t. $$f(x)-f(a)=f'(c_x)(x-a).$$ Since $c_x\underset{x\to a}{\longrightarrow }a$, the claim follow.
Surb
- 55,662
0
Actually, we have more than that:
Let $f$ be a continuous function on an interval $I$, $a\in I$. Suppose $f$ is differentiable on $I\smallsetminus \{a\}$ and $\lim_{x\to a}f'(x)=\ell\in\mathbf R$. Then $f$ is differentiable at $a$ and $f'(a)=\ell$.
Bernard
- 175,478