The way to break out of this circular definition, of course, is to use the delta-epsilon definition of the limit to prove that $x^2$ is continuous at any point. And for any function that we want to evaluate the limit of by simply plugging in the approach value (Think $sin$, $cos$, $ln$, etc. basi...
The epsilon-delta method is only applicable to metric spaces. The topological definition applies to any topological space and reduces to the epsilon-delta definition for metric spaces. en.wikipedia.org/wiki/…
@CyclotomicField What is the point of this comment? This is why I despise this website often. The OP clearly does not know anything about metric spaces, topological spaces, ect. So why reply to the OP with a link to Wikipedia on topological continuity? Some people on this site just want to lowkey say "look at me, I know this more advanced concept, so I am gonna comment with it so everyone can see that I know it". This comment does not help the OP in any way.
@Lauren S Usually, in Calculus I, the course is sloppy, like you said, in that they do not verify details of continuity carefully. However, in advanced calculus (real analysis), you generally prove that many familiar functions are continuous, and then you use those functions as building-blocks to form new continuous functions. For instance, $f(x) = x^2$ is a product of two functions $g(x)\cdot g(x)$ where $g(x) = x$. It is rather "easy" to verify that $g$ is continuous using delta/epsilon. And one goes on to show a product of continuous functions is continuous ...
@NicolasBourbaki the point of my comment is to provide more information so they can further explore the subject. The topological definition of continuity is simpler than the epsilon-delta definition and I wasn't aware that it's considered more advanced. I'd also note that the OP answered their own question so rehashing what they already know seems less useful than providing them with an easier approach. It's exactly what I would want when I asked a question or post an answer. Whatever motivations you've hallucinated I have other than to be helpful are entirely of your own imagining.
@CyclotomicField "The topological definition of continuity is simpler than the epsilon-delta definition." So why is there not a single book in introductory analysis which defines continuity using open sets instead?
@NicolasBourbaki "Introduction to Analysis" by Douglass does. The simplifications brought by the topological definition are a major theme of the text. They're rather passionate about it and I found their reasoning compelling. Epsilon-delta proofs have a reputation for being difficult and that's been my experience with them as well. I don't think I'm alone in that either.
@CyclotomicField No, Douglass does not. I checked the book. He starts chapter 3 with a long section, 3.1, on epsilon-delta and how it is used to define continuity. You know why he starts with epsilon-delta? Because it is easier to learn. Furthermore, it is unavoidable. It is impossible to prove any theorems about differentiable functions using topology alone, you must use epsilon-deltas again anyway.
@NicolasBourbaki cherry-picking one paragraph won't rewrite the book. It's a major theme in the text and nothing you say will change that. You're clearly intent on being angry over nothing so I'll leave you to it.
@CyclotomicField You wrote "topology is easier than delta-epsilon". I asked you for an example. You gave me an example. And your own example of a textbook, not mine, starts with delta-epsilon first. You should look up the definition of cherry-picking.
@NicolasBourbaki you asked for an example of the topological definition being used in an analysis text and when I provided you one you decided that reading a single sentence from a single paragraph somehow made that example disappear. If you want to have an honest discussion about which definition is easier I'm happy to but your dishonesty isn't compelling.
@CyclotomicField I asked, "So why is there not a single book in introductory analysis which defines continuity using open sets instead?" (because according to you, it is easier). You provided Douglass. I looked at the book. And you are incorrect. The book does not define continuity using open sets.
@NicolasBourbaki definition 3.1.2 is right there, just a few words after you stopped reading. The whole of chapter 2 is dedicated to the topological properties of the real numbers as a lead up to the chapter on continuity. The one-point compactification of the reals is given as an exercise and is purely topological. They use the topological definition to prove that continuous functions on a compact set are uniformly continuous. It's impossible to miss if you make even a trivial effort.
@CyclotomicField I am looking at the book again. His first definition for continuity is in terms of deltas and epsilons. The rest of the chapter is all dedicated to deltas and epsilons. According to you "topology is easier". So why is it, that the author uses deltas and epsilons as his first definition? Why did the author not simply start with the topology first completely avoiding any mention of metric spaces? Should I give you a hint?
@CyclotomicField Your claimed that "topology is easier than metrics". The very book that you gave defines continuity with metrics. Why does the book not simply abandon the entire notion of metrics completely and just immediately start with open sets and define continuity as preservation of open sets under pullbacks? Clearly you are wrong and you do not want to admit it.
@CyclotomicField I read your book. It failed my criterion. You just got exposed as being incorrect, that made you upset, and you cannot acknowledge that. This is why my original comment is sitting with 11 upvotes from people who all agree with me and your comments got no upvotes.
@NicolasBourbaki you've definitely made me understand that some people take comments extremely personally. I'll be sure to account for the fragility of the reader in the future.
@CyclotomicField So now you have completely abandoned the original discussion and are pivoting on totally irrelevant. Let me ask you this question. Name me a single intro-level textbook that defines continuity using topology alone without any mention of deltas and epsilons. Your textbook fails that criterion.
@NicolasBourbaki there wasn't a discussion. You decided you wanted to fight instead of talk. I like answering question and I like providing more information to users. You apparently think that's a problem and it ruins the website for you. If you have anything constructive to add now's the time to mention it.
@CyclotomicField So now you abandoned the discussion because you realized that you cannot counter my points? Let me repeat the question again. Why do textbooks even bother to mention deltas and epsilons if they are truly harder than the use of open sets?
@CyclotomicField Now that you conceed that you cannot counter my points let us move on to the next point that you make. Namely, that (in your quote), "I like answering questions and providing information to users". My original comment said that your comments are not helpful. What evidence can you provide that your comments were helpful to the OP?
@NicolasBourbaki what evidence can you provide that they weren't helpful? You've imagined it's so but I have no reason to believe you. I have double the points you have in half the time and it's almost all from answering questions and providing information. If you think you can be more helpful then get on my level.
@CyclotomicField Yes, I can. You provided a comment to the OP. The OP did not tell you it was helpful. You got no up-votes for your comment. "Helpfulness" is judged by community voting. That is the evidence you can use. You got 0 votes. I got 11 votes when I told you, you were being unhelpful. Now, what evidence do you have that your comment to the OP was helpful?
@CyclotomicField For instance, instead of mentioning anything about metrics/topology, which is clearly above the knowledge of the OP (who appears to be either in precalculus or calculus 1), you could have explained how you can fix the circular nature of how continuity is taught. The OP made the correct observation that the way continuity is presented feels very circular. He did say he heard something about delta/epsilons. You could have written an argument using terms he already heard before.
Discussion on answer by Lauren S: Cir…
Discussion on answer by Lauren S: Cir…