Differentiability of the mean value

[jostpuur]

So if a function

<br /> f:[a,b]\to\mathbb{R}<br />

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

<br /> f&#039;(\xi_x) = \frac{f(x)-f(a)}{x-a}<br />

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

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