Decomposable functions were introduced in this article, p. 186.
Given a function $f:I\to \mathbb R$, $f$ is said to be increasing (resp decreasing) at $x$ if, for all $h > 0$ such that $x+h$ belongs to $I$,
$${f(x+h)-f(x)\over h} \geq 0 \quad (resp. \leq 0).$$
$f$ is said to decomposable at $x$ if it is either increasing at $x$, or decreasing at $x$, or $x$ is a STRICT extremum of $f$ in $I$.
Finally, $f$ is locally decomposable in $I$ if, for every $x\in I$, there exists an open interval $J\ni x$ of $I$ such that $f$ is decomposable at $x$ in $J$.
Continuous locally decomposable functions provide a nice change of variable theorem for the KH-integral (actually, the aforementioned article deals with locally decomposable functions at all but countably many $x$).
For example, any function which has a Taylor development of order $n > 0$ at each point is locally decomposable.
I was not able to find a characterization of these functions in term of known concepts like "monotonic", "bounded variation", "ACG" etc. I would be surprised, though, if it were not possible to characterize these functions using known classes of functions.
So, my question is: How to characterize nicely these functions?