Gamma Function is Unique Extension of Factorial

Theorem
Let $f: \R_{>0} \to \R$ be a real function which is positive and continuous.

Let $\ln \circ f$ be convex on $\R_{>0}$.

Let $f$ satisfy the conditions:
 * $f \left({x + 1}\right) = \begin{cases}

1 & : x = 0 \\ x f \left({x}\right) & : x > 0 \end{cases}$

Then:
 * $\forall x \in \R_{>0}: f \left({x}\right) = \Gamma \left({x}\right)$

where $\Gamma \left({x}\right)$ is the Gamma function.