A $\Delta^0_0$, or rudimentary, functions $\Bbb N^k \rightarrow \Bbb N$ is a function whose graph is definable by a bounded formula. Can this class of functions characterized by means of closure conditions, rate of growth, or other (non-syntactic) conditions?
I looked at Odifreddi's Volume II but it was not very helpful. I don't have access to books on bounded arithmetic now.
As explained in this answer https://mathoverflow.net/a/198434, the class is equal to the entire linear time hierarchy in computational complexity theory. Also, it can be characterized by closure property: it's the smallest class containing the initial functions, subtraction and multiplication, closed under composition and bounded minimization, as explained in Rudimantary relation and primitive recursion: A toolbox by Esbelin and More.