Background & Context : The background of the question is an engineering problem.
I want to efficiently represent a set of integers as rounded real valued functions and quickly be able to calculate their partial sums.
For example fonctions describing the number of bits required to store in my previous question here. Would be very practical to just integrate analytically in floating point mathematics rather than loop summing tens or hundreds of values.
I am on the lookout for functions having the property in my own words "Rounding preservant integrability".
So what I do have is an ordered set of integers $\{i_1,i_2,\cdots\}$. I want to be able to find functions $f$ which fulfill the following requirements stated for these sets of integers.
$$i_k = \text{round}(f(k))\\\sum_{k={x_0}}^{x_1}\text{round}(f(x_k)) = \text{round}\left(\int_{x_0}^{x_1}f(x)dx\right), \forall x_0,x_1 \in \mathcal I$$
Which functions can I prove will have this property?
Can polynomials be candidates?
Can this property somehow be enforced or encouraged thanks to mathematical optimization such as regression and regularization?