real analysis - continuity and limit of a function.

Below is the question:

To what degree would the sequence definition of continuity need to be modified in order to be suitable as a definition for the limit of a function?

In other words,if $f$ is a function and if $(x_n)_{n=1}^{\infty}$ is any sequence of domain points such that $(x_n)_{n=1}^{\infty}$ converges to $x_o$,then lim$_{x\to x_o}$$f(x)=L$ iff
$\ldots$ ?

{HERE Sequence definition of continuity is

$f(x_0)$ exists;

$\lim_{x \to x_o} f(x)$ exists; and

$\lim_{x \to x_o} f(x)$ =$f(x_o)$.

}

I cannot understand what should be iff case?Please help...