Over the course of my studies I often encounter phrases in reference material of the type "and this avoids the need for using $\epsilon$, $\delta$ definitions" or "by this we can omit those complicated $\epsilon, \delta$ arguments", etc. In other words performing stunts in order to get around $\epsilon, \delta$. I've seen enough of this to think that it should be categorized as epsilondeltophobia, if you all will permit. Personally, I was thrilled to learn definitions in these terms because it was one of the first rigorous definitions given to me, all in terms of quantifier logic, and it was used for very fundamental things whose real meaning I always wondered about. In the beginning of course I didn't have a clue how to use the language, but I loved it anyways because it was like, "wooow, deep maan". Not to mention that later on, I began to see that all of the higher-order constructions that were built upon $\epsilon, \delta$-objects worked out perfectly, giving me more satisfaction that whoever came up with $\epsilon, \delta$ language knew what they were doing. So I'm not saying that it's not ok to develop an epsilondeltophobia, as we all do naturally in the beginning…but textbooks (some) seem to promote this fear, even some teachers, and this is what I'm not happy about. I think $\epsilon, \delta$ is great.
Question: who thinks likewise? oppositely?
Edit: I don't want this to come off as a pedantic "rigor or death" statement, or as a suggestion that first courses on calculus should always include $\epsilon, \delta$ (although maybe yes in mathematics). I'm just against the predisposal to it in a negative way.
Best Answer
I believe that the pushback against $\epsilon,\delta$ definitions (which unfortunately spills over to pushback against $\epsilon,\delta$ techniques) is entirely justified because $\epsilon,\delta$ definitions arise from the (unfortunately widespread) confusion between a statement being formal and a statement being rigorous.
Consider the formal "definition" of continuity of a function $f$ at a point $a$: $$\forall\epsilon\exists\delta\forall x(0<|x-a|<\delta\rightarrow |f(x)-f(a)|<\epsilon)$$ This is just an objuscated way of stating the informal, but rigorous:
which is logically equivalent to the conceptually clearer, though still informal, though still rigorous:
which is the contrapositive of the, informal and rigorous, intuitive definition of continuity of $f$ at a point $a$:
I strongly believe that the equivalence of the blocked statements and the IDEA that equivalence expresses, which is that we CAN distill an intuitive notion into a rigorous definition, is much more interesting, important, and memorable, than the formal $\epsilon,\delta$ "definition". Furthermore, I can't even bring myself to calling the formal "definition" a definition, since what it expresses is not a description of what it means for a function to be continuous, but a technique (of $\epsilon,\delta$ proofs) for how to check that a function is continuous.
This, in my opinion, is the reason for the pushback against $\epsilon,\delta$ "definition" and arguments: instead of expressing the rigorous idea or concept of continuity, the $\epsilon,\delta$ "definition" only gives a technique for working with continuity, and, when presented as a definition, only obfuscates the meaning of the concept (in a very efficient way, I might add, since the path from the intuitive and meaningful definition to the $\epsilon,\delta$ definition involves taking a contrapositive...).
Finally, I do think that being aware of how to rigorously translate (as above) from the intuitive definition of continuity to the statement of the $\epsilon,\delta$ technique will certainly not hurt, and I suspect could actually help students in using the ($\epsilon,\delta$) technique, especially with the simple functions that arise in Calculus and basic analysis.
(Someone might criticize the above saying that the notion of a ball is confusing in single-variable Calculus. My perhaps controversial response is that there really isn't any good reason not to teach Calculus using $2$ or $3$ variables from day $1$ and that the narrow viewpoint offered by single-variable Calculus obscures more than it simplifies).