Exercise 1, § 2

a) Let $g$ be a continuous real function for $x \geq 0$. Show that if $g$ satisfies the two identities

it also satisfies the identity

b) Deduce from this that if a function $g$ has a continuous derivative for $x \geq 0$ and satisfies the identity

then it is of the form $a(x-\frac{1}{2})$, where $a$ is a constant (remark that $g$ satisfies the analogous identity where $p$ is replaced by $p^n$; let $n$ tend to $+\infty$ and deduce that $g'(x) = \int_0^1 g'(t)\,dt$).

c) Conclude from b) that the $\Gamma$ function is the only function having a continuous derivative for $x > 0$, which satisfies equation (1) of VII, p. 305, and the multiplication formula (12) of VII, p. 318, for a value of $p$.