Defining $x!$ at fractions and when $x<0$ Below is a screenshot from desmos(graph plotting app) . Here it shows that graph is continuous at all $x>0$ and tends to infinity at some values of $x$ and has very abrupt behaviour at $x<0$ . What I read till date is that it's only defined when $x$ is a whole number. Can someone explain what is it showing and how to calculate $x!$ at places except when $x$ is a whole number

 A: This is (almost) the $\Gamma$ function. The $\Gamma$ function is defined as
$$
\Gamma(x)=\int_0^\infty t^{x-1}e^{-t}dt
$$
The $\Gamma$ function has, for positive integers $n$ (or really any complex numbers where it's defined) the properties
$$
\Gamma(1)=1\\
\Gamma(n+1)=n\Gamma(n)
$$
while the factorial has
$$
0!=1\\
(n+1)!=(n+1)\cdot n!
$$
This is almost the same, but not quite. It can be remedied, however. Setting $f(x)=\Gamma(x+1)$ gives
$$
f(0)=1\\
f(n+1)=(n+1)f(n)
$$
which is to say
$$
n!=f(n)=\Gamma(n+1)
$$
And since the $\Gamma$ function is defined for any real (actually complex) apart from the non-positive integers, we can use this to extend the factorial to all real (actually complex) numbers apart from the negative integers. So what you see is the graph of $f$.
Note that the factorial itself is only defined for non-negative integers. The existence of the $\Gamma$ function doesn't change that.
The $\Gamma$ function is actually known to be the only analytic function on the positive real numbers which has $\Gamma(1)=1$, fulfills $\Gamma(x+1)=x\Gamma(x)$, and is superconvex (its logarithm is convex). In this sense it gives the nicest generalisation of the factorial to non-integers.
