What should the limit of a function be at isolated points?

The limit of a real function $f$ at a real number $x$ is usually defined when $x$ is a limit point of the domain of $f$. But what should the limit of a function be at an isolated point? For example, consider a function $f$ defined only on the integers $\mathbb{Z}$. What should the limit of $f$ be at, say, $1$?

As you mentioned yourself (and can be seen in baby Rudin's chapter 4 on continuity), the limit is only defined at limit points of the domain of the function. If the point is an isolated point, you will not be able to satisfy the $\epsilon$-$\delta$ criterion for arbitrarily small values of $\epsilon>0$, unless you allow $\delta=0$; but in that case, the whole definition becomes completely useless.