So if a function(adsbygoogle = window.adsbygoogle || []).push({});

[tex]

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

[/tex]

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

[tex]

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

[/tex]

Sometimes there may be several possible choices for [itex]\xi_x[/itex]. My question is, that if the mapping [itex]x\mapsto \xi_x[/itex] 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?

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# Differentiability of the mean value

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