If $\lim_{x\to x_0}f(x)=f(x_0)$, then $\forall~\epsilon>0$, $\exists~\delta$ such that $|x-x_0|<\delta \implies |f(x)-f(x_0)|<\epsilon$.
This is exactly what it means to be continuous at $x_0$. In Rudin's "Principles of Mathematical Analysis" he offers this as a second definition of continuity.