Let $f\colon\mathbb R\to\mathbb R$ be continuously differentiable and let's say, for simplicity, that $f(0)=0$. Then by mean value theorem it's $f(x)=f'(\xi)\cdot x \,\text{ for some } \xi \in (0, x)$
What I wondered is: What can we tell about the $\xi$ as we change $x$? My intuition says we should at least be able to find some $\xi\equiv \xi(x)$ that varies continuously with respect to $x$.
Or isn't this necessarily the case? Thanks for any ideas.