given $P(n)=e^{-\mu}\frac{\mu^n}{n!}$, under what conditions can I meaningfully write
$$\sum_{n=0}^N P(n)\approx\int_0^N dx P(x) $$
using $x!=\Gamma(x+1)$? It is done nonchalantly in a document I'm reading but I'm not sure what allows us to do it, since the sum is not a Riemann sum and I see no term that might act as the $dx$ in the limit that disappears from the integrand.