The epsilon-delta definition of continuity As we know the epsilon-delta definition of continuity is:
For given $$\varepsilon > 0\  \exists \delta > 0\ \text{s.t. } 0 < |x - x_0| < \delta \implies |f(x) - f(x_0)| < \varepsilon $$
My question: Why wouldn't this work if the implication would be:
For given $$\varepsilon > 0\ \exists \delta > 0\ \text{s.t. } |f(x) - f(x_0)| < \varepsilon  \implies  0 < |x - x_0| < \delta ?$$

Let us suppose that we have some function $f$. Suppose that in some practical application that we need the value of $f$ to be close to some value $y$ with a an error bound that we call $\epsilon$. Then we need $|f-y|<\epsilon$. We control the output of $f$ by varying the $x_0$ a bit. Now suppose that we know the exact value $f(x_0)=y$. But in practice it is difficult (probably impossible) to get $x=x_0$ on the nose. So we need to know that If we can control $x$ sufficiently well our error $|f-y|$ will not exceed epsilon, no matter what epsilon we choose. We also need to know how tight that control needs to be. The tightness of the control of $x$ to guarantee that we can control $f$ is exactly $\delta$. If the error in $x$ exceeds $\delta$ we might be Ok, but we also might be in trouble.   

What you want is that $f(x)$ is close to $f(y)$ whenever $x$ is close to $y$. You do not really care about whether $f(x)$ is close to $f(y)$ if $x$ and $y$ are far away. 

