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Suppose $f:\mathbb{R} \to \mathbb{R}$ is differentiable. Fix $c \in \mathbb{R}$ and recall the MVT

$\frac{f(x) - f(c)}{x-c} = f'(\delta_x)$ where $\delta_x \in (c,x)$.

What can we say about the "function" $\delta:\mathbb{R}_{x>c} \to \mathbb{R}?$

This is broad, I guess, but some questions which come to mind:

  1. Suppose $f'$ is injective. Then we can remove the quotations from function. Is $\delta_x$ continuous? Differentiable? How does imposing certain conditions affect its behaviour(say $C^{\infty})$? For some functions (which don't meet the injectivity requirement) it can be made discontinuous. How discontinuous can it be?

  2. Suppose $f'$ is not injective; in this case, $\delta_x$ may be multivalued. In particular, it is a set of values. Are there circumstances in which this "set" is ever useful? For example, a reasonable sounding conjecture is that if $\delta_x$ is a set with positive Lebesgue measure, $f$ is linear almost everywhere in $(c,x)$.

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  • If $f\in C^1$ and $f'$ is injective, then $f'$ admits an inverse and so we can express $\delta(x)=(f')^{-1}\left(\frac{f(x)-f(c)}{x-c}\right)$, which is continuous and differentiable. – Reveillark Sep 09 '16 at 00:54

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