Continuity states that for every $\displaystyle \epsilon > 0$, there exists a $\displaystyle \delta > 0$ such that $\displaystyle |x - x_{0}| < \delta \implies |f(x) - f(x_{0})| < \epsilon$. So we pick an arbitrary $\displaystyle \epsilon$. That is, we simply say: Let $\displaystyle \epsilon > 0$, then we solve for $\displaystyle \delta$.