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) ?