Given a positive integer $ n $ , let $ S\_{b}(n) $ the set of functions $ f $ fulfilling the following conditions : 


1) $ f $ is continuous, positive and increasing on $(n,+\infty) $ 


2) for all $ x>n $ , the interval $I\_{f}(x) : =[x, x+f(x)] $ contains at least one prime number


3) the number of primes in $ I\_{f}(x) $ remains bounded as $ x $ tend to $+ \infty $ .


Is $ S\_{b}(n) $ closed under both finite sums and products ? Have similar spaces of functions been studied in prime number theory so far ? Any references (not under a paywall, please) ? 



Space of functions f such that the number of primes in $ [x, x+f(x)] $ remains bounded
There is no such function $f(x)$. First, the prime number theorem implies that the average gap between two primes $p\_{n}$ and $p\_{n+1}$ is about $\log p\_{n}$ and for this reason, if $f(x)$ is smaller than $\log(x)$, then your condition (2) will fail.


On the other hand, Theorem 3.2 from Maynard's paper "[Dense clusters of primes in subsets](https://arxiv.org/abs/1405.2593)" (published in Compositio Mathematica in 2016) states that for any $x$ and $y$ there are (a lot) of integers $x\_{0} \in [x,2x]$ so that
$$ \pi(x\_{0} + y) - \pi(x\_{0}) \gg \log y. $$
It follows from this that if $f(x) \to \infty$, then there are $x$ so that the interval $[x, x+f(x)]$ contains arbitrarily many primes, violating your condition (3).


