True/False: f'(a) Exists if f(x) is Continuous, Limit of f'(x) is c at x->a

[tsuwal]

Homework Statement

True/False:f(x) is continous, limit of f'(x) as x->a is c, then f'(a) EXISTS equals c

Homework Equations

The Attempt at a Solution

I know that if f'(a) exists the statement is true, but is it true that based on that information f'(a) exists?

Homework Statement

Homework Equations

The Attempt at a Solution

 

[jbunniii]

Look at the definition of the derivative. $f'(a) = c$ means that
$$\lim_{x \rightarrow a}\frac{f(x) - f(a)}{x - a} = c$$
Try applying the mean value theorem to
$$\frac{f(x) - f(a)}{x - a}$$
and see if you can conclude anything.

 

[tsuwal]

yes, my teacher explained t that way but the last part of the demonstracion when he uses some theorem about the limit of compound functions with csi(x) is really confusing..

 

[jbunniii]

Suppose $x > a$. Does the mean value theorem apply to $f$ on the interval $[a, x]$? If so, what does it say?