Name for function that is Lipschitz continuous over partitioning of input space

Question

Let $f: X \to \mathbb R$ and $(X,d)$ be a metric space. Let $P=\{P_1,P_2,\dotsc\}$ be a countable partitioning of $X$. I would like to assume that $f$ is Lipschitz continuous on $(P_i,d)$ for all $P_i \in P$, but not necessarily over $(X,d)$. Is there a standard word or phrase that describes this sort of function?

Answer 1

Following up on my yesterday's comment . . .

There are two well-known (in classical real analysis) similar notions in S. Saks Theory of the Integral (1937), VBG (generalized bounded variation, p. 221) and ACG (generalized absolutely continuous, p. 223). A function $f:[a,b] \rightarrow {\mathbb R}$ is VBG if there exists a countable partition of $[a,b]$ such that the restriction of $f$ to each set in this partition has bounded variation on that set, and similarly for ACG. I haven't tried to track down the origin of these notions, but I doubt it would be very difficult to do so if there is any interest.

Anyway, two relatively recent books make use of the more general notion "$f:E \rightarrow {\mathbb R}$ is PG", where $E \subseteq {\mathbb R}$ and P is some property that $f$ and any of its restrictions can have (e.g. bounded variation, absolute continuity, Lipschitz, etc.) --- Real Functions – Current Topics by Vasile Ene (see Definition 1.9.2, and see LG for P = Lipschitz on p. 58) and Theory of Differentiation by Krishna M. Garg (see Section 1.3 on pp. 15-16).

However, I think the use of "generalized" in this context is not likely to be all that well known (or guessable) to those not steeped in a background of classical real analysis, and I suggest maybe using something like $\sigma$-Lipschitz function if you have to refer to the notion a lot and want a short suggestive term. The idea is that what you want is a countable union of Lipschitz functions, the union being over the ordered pairs that define the functions.

I also saw the term "*countably m rectifiable" used in Geometric Measure Theory by Herbert Federer (3.2.14 on p. 251) for a set in ${\mathbb R}^{m}$ that is a countable union of images of Lipschitz functions, but not the idea of a function being a countable union of Lipschitz functions.