I came across the following question when I was dealing with size-distribution of particles in a gas and was trying to formally convert the binned data to a continous size-distribution.
Prior to measurements I divided the measuring range of particle sizes $x$ in $k$ subintervals $[x'_{k},x'_{k+1}]$ and bin sizes $\Delta x_k$. Every particle within the k-th subinterval is considered to be of size $x_k$. The total amount of particles detected over the measuring range is given by $N_{tot}$. Consequently it is
$N_{tot} = \sum_k N(x_k) := \sum_k N_k = \sum_k \frac{N_k}{\Delta x_k}\Delta x_k$
where, $N(x_k) = N_k$ is the amount counted within the k-th bin. Since bin sizes are not normalized, the particles counted within a bin is, per construction, dependent on bin width and $\frac{N_k}{\Delta x_k}$ represents a normalized view.
My question is: How do I - formally correct - turn this discrete distribution into a continous one, as given below?
$\int_0^{\infty} \frac{\partial N(x)}{\partial x} \,\operatorname{d}x = \int_0^{\infty} \, \operatorname{d}N(x) = N_\text{tot}$
My approach was considering the limit of $k \to \infty$ or in other words $\Delta x_k \to 0$, which however brought me to a similiar expression, but left the $N_k$ somehow untouched
$\int_0^{\infty} \frac{N_k}{\partial x} \,\operatorname{d}x$