Epsilon-delta definition of a continuous function

The definition for a function being continuous at a point is obviously well known: Given $\displaystyle \epsilon > 0$ there exists $\displaystyle \delta > 0$ such that $\displaystyle |x - x_{0}| < \delta \implies |f(x)-f(x_{0})| <\epsilon$

But assume we use the following "definition"; Given $\displaystyle \delta > 0$ there exists $\displaystyle \epsilon > 0$ such that $\displaystyle |f(x)-f(x_{0})| <\epsilon \implies |x - x_{0}| < \delta$.

Where does this fail in implying continuity, how is this insufficient to showing that the implication:if $\displaystyle \lim_{n \to \infty} x_{n} = x_{0}$, then $\displaystyle \lim_{n \to \infty}f(x_{n}) = f(x_{0})$ holds (which is the sequential definition of continuity)?