# Functions with “gradients of bounded variation”

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.

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I'm mildly curious about this myself. Where have you looked? My best guess would be Walter Rudin's Introduction to Real Analysis possibly as an exercise. Your problem seems to be the classic problem of wanting to search for math but not being able to. Recently new technologies have been developed to help with this, e.g. latexsearch.com – David White Jun 28 '11 at 16:53