I've become a bit stuck on a problem I'm working on. I have a sum of a piecewise function defined as follows: $$ f(x) = \sum_{n=0}^{N} \begin{cases} \delta & x\leq g(n\delta) \\ \delta \frac{g(n\delta)^3}{x^5} & x\geq g(n\delta) \end{cases} $$ Here, $g(n\delta)$ is a function and $\delta$ is a small, constant increment. Thus as $n$ gets large and $\delta $ becomes infinitesimally small, the sum begins to look like an integral.
I feel like I should be able to write this as a collection of definite integrals, but I'm not sure where to begin, nor how to deal with the fact that the limits are variable.
Any clues on what I should do here?
Thanks, Hugh
EDIT: I think the real issue I'm struggling with is how to handle a domain that varies with the summation index, e.g.: $$\sum_{i=0}^{N} \left(Q \textrm{ for: } x \leq x_i\right)$$ $Q$ may be a constant, but the domain varies with $i$, and is therefore included in the summation.