Differentiability of the mean value

[jostpuur]

So if a function

$$f:[a,b]\to\mathbb{R}$$

is differentiable, then then for each $x\in [a,b]$ there exists $\xi_x \in [a,x]$ so that

$$f'(\xi_x) = \frac{f(x)-f(a)}{x-a}$$

Sometimes there may be several possible choices for $\xi_x$. My question is, that if the mapping $x\mapsto \xi_x$ is chosen so that it is continuous, is it always also differentiable? In other words, does the limit

[tex] \lim_{h\to 0}\frac{\xi_{x+h}-\xi_x}{h}<br /> [/itex]<br /> <br /> exist?[/tex]

 

[MathematicalPhysicist]

what you wrote is correct when: f'(e)(x-a)=f(x)-f(a), cause of the choice of x=a.
for your question you mean the function at the points ksi_x+h and at ksi_x.

well this is ofcourse correct when f is differentaibale continuous.
btw, it's enough to assume that it's differentiable in (a,b) and continuous in [a,b].