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If, for some function $f$ and some value $p\ne 0$, $$\forall x, f(x)=f(x-p),$$ then $f$ is periodic and $p$, if $>0$ and minimal, is $f$'s period. If, instead, for some values $u\ne 0$ and $v$, $$\forall x, f(x)=f(x-u)+v,$$ then what is the correct adjective to describe $f$, and what are the correct terms for $u$ and $v$ in relation to this $f$?

The domain is a subset of $\mathbb{R}$. The codomain might, but need not, be a subset of $\mathbb{R}$; just any group.


I have some terminology in mind, but have come to doubt my term for $v$ because (w.r.t. functions) that term is used to denote something else. So I seek other opinions. In the fullness of time I will edit this OP to state the terms I use at the moment, but I want to avoid the situation where people answer "yes, that's OK" (which doesn't me any further) and others are put off answering because this question looks to them as if it has a satisfactory answer (which doesn't get future readers any further than I am).

EDIT: A staircase function is piecewise constant on intervals of length $u$, so I feel that dxiv's suggestion of "staircase-like function" suggests that the function might be something like that -- that is not my intention. Anyway, the terminology I already knew is that from combinatorial game theory: "arithmetico-periodic", "period" and "saltus". "Period" is of course problematic because of the more specific sense, which is well established. And "saltus" is problematic because of its sense "jump discontinuity".

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  • $\begingroup$ Such an $f$ is a periodically perturbed linear function. $\endgroup$ – Christian Blatter Dec 14 '17 at 10:30
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Not an authoritative answer by any stretch, but I'd call it maybe "staircase-like function", or perhaps "linearly augmented periodic function". The latter coming from the observation that $f(ux) - v x$ is in fact periodic: $$\require{cancel}\;f\big(u(x+1)\big) - v(x+1) = f(ux+u)-vx-v = f(ux)+\bcancel{v}-vx-\bcancel{v}=f(ux)-vx\,$$

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