We apply the Mean value theorem to a real analytic function $f$ (defined on $\mathbb R$) in the interval $(u,a)$ such that $u<a$ and $f(u)=0$ to find a $c\in(u,a)$ such that: the expression $f(a)/(a-u)$ is the slope of the chord of the graph of $f$, and $f'(c)$ is the slope of the tangent line to the curve at the point $(c,f'(c))$. That is we have:
$f'(c)=f(a)/(a-u)$ (*)
I am interested in the set of all $c$ satisfying the equation (*). I want to classify all these values by using an equivalence relation. However, I am not able for formulate this relation mathematically.
As long as $u$ and $a$ are fixed the set $$S:=\left\{c\in \ ]u,a[\ \biggm|\ f'(c)(a-u)=f(a)-f(u)\right\}$$ is just a set, and the MVT guarantees that $S$ is not empty. You may declare any two numbers $x, \> x'\in \ ]u,a[\ $ as equivalent if the characteristic function $1_S$ assumes the same value on $x$ and $x'$, but this produces no new insight.