Given limit f(x) = L at x=c , does the function f(x) "always" need to be defined in the close neighborhood of c?
Yes and no. Any function $f:A\to B$ of course has to be defined on all of $A$, otherwise it is not a function. However, you may defined $f:[0,1] \to \mathbb{R}$ and still pick $c=0$ and, say, $\delta = 2$. This does not mean that $f$ must be defined on the interval $(-2,0)$, however: The definition says more precisely that for $x\in A$ satisfying $c-\delta<x<c+\delta$, $f(x)$ has to be $\epsilon$-close to $f(c)$. With the quantifier $x\in A$, we do not run into any problems.