Alternative definition of factorial

Question

Suppose $$f(x)=x!$$ for all $x$ in $N$,and $f$ is also continuous in $x>0$,does this mean $f(x)=\Pi (x)$ for all $x>0$?

Answer 1

Do you see why the following question has a negative answer?

If $f$ is continuous on $(0,+\infty)$ and $f(n)=n$ for all natural numbers $n$, must $f(x)=x$ for all $x$?

Of course not, because $f(x)=\cos(2\pi x)x$ has exactly the same property, but surely isn't equal to $x$ everywhere.

Your question has a negative answer for the same reason: knowing only what it does on integers and nothing else (beyond continuity) is not going to make your function unique.

Answer 2

Strictly speaking, the question doesn't make sense, as the factorial operator is only defined for natural numbers. However: You might want to look up the Gamma function.

Answer 3

There are infinitely many continuous functions $\mathbb{R} \rightarrow \mathbb{R}$ that satisfy $f(x)=x! \space \forall x \in \mathbb{N}$.

Start with one such function $g(x)$ and then add any continuous function that is zero at all integers. For example you can create an infinite family $g_n(x)$ as follows:

$g_n(x) = g(x) + \sin (n \pi x)$