Is this function defined through mean value theorem smooth enough? Through mean value theorem we know that for a differentiable function $f$ on reals, for any distinct $x, y$ in $ \Bbb R$, there exists a $c$ in $(x,y)$ such that $$\frac{f(x) - f(y)}{x-y} = f'(c)$$
I was wondering, if we fix this $x$ and vary only $y$ (thus, in turn varying the value of $c$), we might be able to define a function $f_x$ which returns a possible value of $c$ for each $y$ . My question is, how smooth can we get this function to be ? Please let me know if there any results of this sort that I am unaware of.
(Note: As it is possible that a single y can give us different values of $c$ and we are just picking one of them, what I am asking is how nice can we get this $f_x$ to be and not that all possible definitions should be nice)
I am not sure I can even prove that there is a continuous way to define $f_x$ when $f$ is only given to be differentiable once. However, I suspect that if $f$ is twice differentiable, it might be possible to achieve something on this, using inverse function theorem.
 A: Not a full answer, but maybe this definition will help proving continuity:
We search a mapping $$c_x: y \mapsto c_x(y)$$ such that $$ \frac{f(x)-f(y)}{x-y} = f'(c_x(y))$$
We assume for simplicity that $x<y$.
Mimicking the proofs of the Mean value theorem and Rolle's theorem we define  mappings $$g_y: [x,y] \to \mathbb{R} \\
t \mapsto f(t) - \frac{f(x)-f(y)}{x-y}t$$
Now, the proof of rolle's theorem works by finding either a minimum or maximum of this function in the interior of the interval, where this depends on the behaviour at the endpoints.
(I.e. if the minimum is taken at the endpoints, then a maximum is chosen, and vice -versa, if both are taken at the endpoints, the function is constant on the interval)
Now, if $g_y$ is not constant, we thus can define
$$ c_x(y) = \max (m(y),M(y))$$
Where
$$ m(y) = \inf \{ t \in (x,y) | g_y(t)D(y) = \underset{t \in [x,y]}{\min} g_y(t) D(y)\} \\
   M(y) = \inf \{ t \in (x,y) | g_y(t)D(y) = \underset{t \in [x,y]}{\max} g_y(t) D(y)\} \\
   D(y) = \underset{ t \in [x,y]}{\max} g_y(t) - \underset{ t \in [x,y]}{\min} g_y(t)$$
This definition will yield an unique point in the interior of the intervall $[x,y]$ as long as $g_y$ is not constant on this interval, which might need another modification.
