Why is Euler’s Gamma function the “best” extension of the factorial function to the reals?

The Bohr–Mollerup theorem shows that the gamma function is the only function that satisfies the properties

• $f(1)=1$;
• $f(x+1)=xf(x)$ for every $x\geq 0$;
• $\log f$ is a convex function.

The condition of log-convexity is particularly important when one wants to prove various inequalities for the gamma function.

By the way, the gamma function is not the only meromorphic function satisfying
$$f(z+1)=z f(z),\qquad f(1)=1,$$
with no zeroes and no poles other than the points $z=-n$, $n=0,1,2\dots$. There is a whole family of such functions, which, in general, have the form
$$f(z)=\exp{(-g(z))}\frac{1}{z\prod\limits_{m=1}^{\infty} \left(1+\frac{z}{m}\right)e^{-z/m}},$$
where $g(z)$ is an entire function such that
$$g(z+1)-g(z)=\gamma+2k\pi i,\quad k\in\mathbb Z,$$
($\gamma$ is Euler’s constant). The gamma function corresponds to the simplest choice
$g(z)=\gamma z$.

Edit: corrected index in the product.