Continuity

The limit of a function $f$ at $x_0$ is $L$ iff ${\lim\limits_{x \to x_0}} f (x) = L$ iff

$$\forall \epsilon > 0, \exists \delta >0 : \forall x : 0 < |x - x_0| < \delta \to |f(x) - L| < \epsilon$$

$f$ does not need to exist at $x_0$ to define the limit. However, for the function to be continuous, the $\lim\limits_{x \to x_0} f(x) = f(x_0)$.