You are right that, by the traditional definition of $x!$, it is only defined at the integers. However, we can analytically continue the factorial function to all real numbers (except, as Peter notes in the comments, for all negative integers and $0$). This continuation (along with a shift) is known as the Gamma Function.
However, note that there are other continuations of the factorial function. The one pictured here is the Gamma Function. Here are some other continuations which interlope $x!$.