I was wondering (or mind-wandering) about a function described as:
For any given $x_1$, $x_2$, $x_m = \frac {x_1+x_2} {2}$, $f(x_m) = f(x_1) + a (f(x_2) - f(x_1)) , a \in ]0;1[$
For example, $f(x_m)$ is $\frac 23$ of the distance between $f(x_1)$ and $f(x_2)$.
I can intuitively see that such a function isn't consistent if $a \neq \frac 1 2$ but how to prove it mathematically.