My question is mostly: is there a name for this kind of things. I am mostly interested by finding book or articles about what follows, but without even a word or a name, it is quite hard to search for information.
Let $F$ be the smallest class of functions $f(x_1,\dots,x_n)$ which contains the functions: -real constants -projection on one component (that is, $x_i$ for some $1\le i\le n$, -$\lfloor f\rfloor$, for some $f\in F$, -$f_1+f_2$, for $f_1,f_2\in F$ -$cf$ for some $c\in\mathbb R$ and $f\in F$
As stated by Dmitri Zaitsev, the functions in $F$ are piecewise affine functions. But this description is too general, there are piecewise affine function which lacks the periodicity provided by the floor function. Therefore, I would like a name for $F$, or at least, to know which kind of property is satisfied by functions of $F$.
(Of course, those functions are interpreted on $\mathbb R$, but if it helps it could be $\mathbb Q$, as I don't know any other field where $\lfloor\rfloor$ is defined, apart from $\mathbb Z$ where the question becomes trivial.)