It occured to me that while I think I have a grasp of Riemann integration, I do not know of theorems regarding the rates of convergence for such integrals. So I would appreciate it if anyone would point some theorems in the subject or give me reading material on the subject.
For example, given the uniform partition of $[a,b]$ to $n$ equal length intervals, can we give a condition a concrete condition on $n$ such that $\vert S(P_n;f)-\int_a^b f\vert<\epsilon$? I am of course assuming one would have to lose the generality of $f$ for this estimate.
The usual proof that continuous functions are Riemann integrable on compact intervals gives the fairly explicit bound $ n > K|a - b|/\epsilon $ whenever $ f $ is Lipschitz continuous with constant $ K $, which will certainly be true if $ f $ is of class $ C^1 $, for instance. If $ f $ is not Lipschitz continuous, getting such clean bounds on $ n $ becomes more difficult.