The classical way the $\epsilon$-$\delta$ definition for continuity (limits) proceeds is by starting with a given $\epsilon>0$, and our ability to then find a number $\delta>0$, such that when $|x-x_{0}|<\delta$, we get $|y-y_{0}|<\epsilon$.

My question is: why cannot we do exactly the same definition but starting from a given $\delta$ (instead of starting with a given $\epsilon$)?

That is, why not say: given $\delta>0$, we can find a number $\epsilon>0$, such that when $|x-x_{0}|<\delta$ we get $|y-y_{0}|<\epsilon$ ?

Note: this question is different from other questions posted regarding the direction of implication of the definition (e.g. this post)


merged by Jyrki Lahtonen Nov 3 '16 at 7:21

This question was merged with Slightly changing the formal definition of continuity of $f: \mathbb{R} \to \mathbb{R}$? because it is an exact duplicate of that question.