$X$ is a continuous random variable and $F_X(x)$ is its differentiable cumulative density function. I'm considering what happens if I discretize $X$ to intervals of length $\Delta x$ (i.e. a discrete $Y$ such that $P(Y=y)=F_X\left(y+\Delta x\right)-F_X(y)$ for a partition of the support of $X$). Specifically, the Shannon information of such a random variable is $$S(Y)=-\sum_{y}\left(F_X\left(y+\Delta x\right)-F_X(y)\right)\log_2\left(F_X\left(y+\Delta x\right)-F_X(y)\right)$$ Intuitively, if we let $\Delta x\rightarrow 0$, $S(Y)\rightarrow S(X)$*. However, evaluating such a limit is proving rather difficult because of that differential in the logarithm**. I've explored what I can find (and understand) of geometric integration and hyperreal analysis, but the solution evades me.
*Maybe that's not actually true.
**Yes, I am aware of the concept of differential entropy, but both my own attempts at this and a paper I recently skimmed (and now cannot find, of course) seems to indicate that differential entropy is not the continuous analog of Shannon entropy.