Being that it is the end of the semester I have an Analysis final coming up in a few days. On previous tests my professor has asked for a proof that a certain function was continuous explicitly requiring that the proof use the $\varepsilon$-$\delta$ method of continuity.

The thing is, like many students, I really don't like the $\varepsilon$-$\delta$ method. I much prefer to prove that the pre-image of every open set on the domain is open, I find it more intuitive and it was what I was using for a while before questions requiring the $\varepsilon$-$\delta$ method started popping up.

In a proof by example, Hagen von Eitzen pointed out that there are certainly many cases where it is a better idea to use $\varepsilon$-$\delta$ proofs over the topological ones. However there are still cases where I find it easier to the topological definition.

My question is it there some way I could convert a proof using the pre-image of open sets definition to a $\varepsilon$-$\delta$ format? Or use my intuitions of one proof method to generate proofs in the other? My main issue with rewriting my proofs is that the pre-image definition deals with continuity of an interval rather than the continuity of a single point, while the $\varepsilon$-$\delta$ definiton defines continuity at a single point.

I understand that $\varepsilon$-$\delta$ proofs are an important skill to have, and that my professor probably has good intentions in requiring them. I plan to practice my $\varepsilon$-$\delta$ skills before the test, but I still feel nervous I might blank on the test and I would like to have a back-up plan for these types of questions.

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    $\begingroup$ Does it help to realize that the $\epsilon-\delta$ definition is simply the specialization of the open pre-image of open sets definition to a metric topology? Given an epsilon-ball in the image, one finds a delta-ball in the preimage such that.... $\endgroup$ – hardmath Dec 14 '17 at 7:24
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    $\begingroup$ Could you give an example of how you would write a pre-image-of-open-is-open proof of, say, the continuity of $\Bbb R\to\Bbb R$, $x\mapsto x^3+7x$? $\endgroup$ – Hagen von Eitzen Dec 14 '17 at 7:45
  • $\begingroup$ Aah! the curse of topology... it looks deceptively simple! As von Eitzen commented, how else to prove that $x^3+7x$ (or $x^3-7x$) is continuous without some sort of $\epsilon-\delta$ proof? $\endgroup$ – Chrystomath Dec 14 '17 at 7:59
  • $\begingroup$ @MMS You can use theorems such as the sum and products of continuous functions are continuous. It only remains to prove that $x\mapsto x$ and $x\mapsto 7$ are continuous. $\endgroup$ – Gribouillis Dec 14 '17 at 8:02
  • $\begingroup$ I noticed that often people were hating $\epsilon-\delta$ because they were puzzled about foreseeing in advance in how much that have to cut their $\epsilon$ or how much precise they need their $\delta$ so that in the end they get $|foo-bar|<\epsilon$. As soon as you realize though that rough majorations suffice and you need only $|foo-bar|<K\epsilon$ for some constant $K$ then the pressure on this kind of proofs just goes poof. Maybe did you experienced this kind of blocking towards the $\epsilon-\delta$ proofs. $\endgroup$ – zwim Dec 14 '17 at 8:05

I think the simplest proofs are the ones that use sequential continuity, that is to say $f$ is continuous if and only if $$\lim x_n = a \quad\Longrightarrow\quad \lim f(x_n) = f(a)$$

These proofs are easy to design and they are excellent starting point to create a $\epsilon - \delta$ proof.

  • $\begingroup$ Thanks for taking the time to write this up, however I don't really think that this answers the question I asked. This is a way of generating an $\varepsilon$-$\delta$ proof, but doesn't concern itself at all with the topological definition. $\endgroup$ – Sriotchilism O'Zaic Dec 14 '17 at 21:02
  • $\begingroup$ As @hardmath pointed out, if you are able to prove that the preimage of any open set $U$ is open, then you are able to prove that the preimage of $B(f(a), \varepsilon)$ is open, then you are able to prove that $f^{-1}(B(f(a), \varepsilon))$ contains a non empty ball $B(a, \delta)$, then you are able to give a $\epsilon -\delta$ proof. We need you to give concrete examples to go farther. $\endgroup$ – Gribouillis Dec 14 '17 at 21:09

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