We know that the gamma function is
$$\Gamma(n+1)=n!=\int_0^\infty e^{-x}x^n dx=I_n$$
It can be easily shown, via integration by parts, that
$$I_n=nI_{n-1}$$
which confirms its suitability as a continuous function to represent the factorial function.
Question
Is it possible to construct another integral $J_n$ or continuous function $f_n$ that would have this property, i.e. $J_n=nJ_{n-1}$ or $f(n)=nf(n-1)$, or show that no such other integral or function exists?
You could take $g(x) \Gamma(x)$ for any function $g$ that is periodic with period $1$.