Dear all,
I would like to know whether the following concept is one that is commonly studied, or has a name, or if there are any textbooks that make reference to it:
We say that a function $f:[a,b] \to \mathbb{R}^n$ satisfies "property X" if
$\exists M > 0$ such that for any partition ${a = t_0 < t_1 < \ldots < t_n = b }$ with $n \geq 2$,
$\sum_{i=0}^{n-2} \left| \frac{f(t_{i+2}) - f(t_{i+1})}{t_{i+2} - t_{i+1}} - \frac{f(t_{i+1}) - f(t_i)}{t_{i+1} - t_i} \right| < M.$
I am interested in knowing whether "property X" has a standard name and if there are any textbooks which discuss it.
(If $f$ is differentiable, then satisfying property X is, I think, equivalent to $f'$ having bounded variation. But functions which are not everywhere differentiable can still satisfy property X.)
Many thanks indeed. Julian.
