weaker condition for continuity.

Let $\displaystyle f:\mathbb{R} \rightarrow \mathbb{R}, \, a\in S \subset \mathbb{R}$.Suppose $\displaystyle f$ has the property that:

$\displaystyle x_n \in S, \, x_n \rightarrow a \Rightarrow (f(x_n))$ is convergent. Prove that $\displaystyle f$ is continuous at $\displaystyle a$.

my approach:

Let $\displaystyle (x_n)$ and $\displaystyle (y_n)$ be two sequences in $\displaystyle S$ with $\displaystyle x_n \rightarrow a, \, y_n \rightarrow a$. let $\displaystyle f(x_n) \rightarrow L_1$ and $\displaystyle f(y_n) \rightarrow L_2$. I need to prove that $\displaystyle L_1=L_2$. how do i go about it?