I was wondering because of this: Trick to find if number is composite or prime
Is there any formal method to convert a discrete function to a continuous function. For example take $n!$, how was the gamma function discovered? Is there a general procedure to get the continuous function (which mimics a discrete function)?
In general, there any many ways for extending a function $f:\mathbb{Z}\to\mathbb{R}$ in such a way that $g$ is a continuous function and $g_{|\mathbb{Z}}\equiv f$. For the $\Gamma$ function, the uniqueness of the extension follows from requiring that for any $z\in\mathbb{R}^+$, $g(z+1)=z\, g(z)$ (the same functional identity satisfied by $f$ on $\mathbb{N}$) and $\log g$ is a convex function: see the Bohr-Mollerup theorem.
Depending on how the discrete function is defined, there might be simple ways to extend it to a continuous function on $\mathbb R$, e.g. if $f: \mathbb N\to\mathbb R$ is defined as the constant $65$, it's quite obvious to consider the function that is constant $65$ on $\mathbb R$ as an extension, but $65\sin(\pi x)$ is also a choice.
In some cases the function has other properties that helps us choose, but different properties also means that there are no general way of finding a valuable extension.
If the function satifies a functional equation like $n! = n(n-1)!$ that can sometimes be used as a starting point for finding some functions that has matches the discrete function.
