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}
[/itex]

exist?

[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].