A big part of introductory real analysis courses is getting intuition for the $\epsilon-\delta\,$ proofs. For example, these types of proofs come up a lot when studying differentiation, continuity, and integration. Only later is the notion of open and closed sets introduced. Why not just introduce continuity in terms of open sets first (E.g. it would be a better visual representation)? It seems that the $\epsilon-\delta$ definition would be more understandable if a student is first exposed to the open set characterization.
Why do introductory real analysis courses teach bottom up?
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6I agree; time for my soap box. I hate the definition of continuity normally given, because it is one of the least motivated definitions I have heard outside of category theory. Why do they define continuity in terms of $\epsilon,\delta$? It's easy to check that a function satisfies this. Why shouldn't they? Because it takes people a long time to understand what continuous functions *are* if you define them that way, and at first continuity seems absolutely worthless. Personally, I'd start by defining convergence of sequences and define continuity in terms of commuting with limits of sequences. – 2012-02-26
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0@AlexBecker I'm fairly sure (but not absolutely certain) that what you suggest is how I was first taught continuity, and I agree that it feels much more intuitive than the explicit quantified laden epsilon-delta definition – 2012-02-26
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10I totally disagree with Alex Becker's comment. The basic notion is *continuity*: A function is continuous at a point $x_0$ if inputting an $x$ "near" $x_0$ produces a value $f(x)$ "near" $f(x_0)$. The difficult part is that the quantors work *backwards*: For any $\epsilon > 0$ there is a $\delta > 0$ such that, and so on. This difficulty remains when you start with sequences. Only you have $2^{{\rm aleph}_?}$ sequences to check until you have proven a single instance of "$f$ is continuous at $x_0$". – 2012-02-26
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0As for introducing continuity in terms of open sets, I believe that Joel Feinstein has been experimenting with something that sounds similar: see http://explainingmaths.wordpress.com – 2012-02-26
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7One nice thing about the $\epsilon-\delta$ version is that it leads naturally to the discussion of uniform continuity, which then leads to stuff about convergence of sequences of functions. And practically speaking, $\epsilon-\delta$ proofs are the bread-and-butter of a lot of analysis, so it is probably a good thing to expose people to it. And to be honest, I am not sure the topology definition is easier to visualise... – 2012-02-26
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3@ChristianBlatter My point is that I feel commuting with limits explains why continuity is a *useful* property for functions to have, more so than the $\epsilon-\delta$ definition. – 2012-02-26
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0@WillieWong but I would argue that the definition via sequences is a useful first step when first doing proofs that sums, products etc of continuous functions are continuous - a structuralist approach to analysis, as it were. I agree that epsilon-delta is more natural if you want to segue into uniform continuity. – 2012-02-26
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0@YemonChoi my comment was more about the OP's question, which seemed to be about the topology definition. From my point of view, the $\epsilon-\delta$ and the sequences definition are more or less equivalent. And also, don't most students taking real analysis have had a course in calculus first? (Or do calculus courses nowadays start immediately in the epsilon-deltas?) – 2012-02-26
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6It seems clear to me from his question that "Jones Indiana" has never taught students at this level before. – 2012-02-26
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2I don't think the difficulty with limits is due to the definition itself. The definition, once understood, is simply a numerical realization of the concept that for any change in the output, no matter how small, one can find a tolerance level in the domain that induces it. Conceptually, I think this is very clear and has always been clear to me. The difficulty arises in the various tortures one must endure in order to pick an appropriate epsilon that verifies this. This is something that I struggled with from the beginning - and continue to struggle with. – 2012-02-26
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1@GEdgar My interpretation of your comment is that it would be nearly impossible to get third/fourth year math students to understand topological concepts like continuity. In reality, what Jones Indiana is proposing could easily be a quick homework exercise. Instead of "open set" just use "open interval" which they understand already. For goodness sake it is $\mathbb{R}$ and say a continuous function is one that has the property that preimages of open intervals are open sets. Then make them check that using $\epsilon$-$\delta$ you get exactly the same thing. – 2012-02-26
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0@AlexBecker: but really you're passing the buck -- after all, aren't you going to define convergence of sequences by an $\epsilon - N$ definition? – 2012-02-27
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0@Hurkyl Yes, but that one seems more basic and and intuitive to me, as you're "trying to get close to a single point". – 2012-02-27
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0@AlexBecker, and how are you going to actually prove that a specific function converges? – 2012-02-27
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0@Mariano first get the function $x\mapsto x$, then use results on products and sums of limits to get polynomials. For other functions, sandwich arguments might work. – 2012-02-27
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0@MarianoSuárez-Alvarez I would definitely use $\epsilon-\delta$ after proving it equivalent. But all I'm saying is that isn't the first definition I'd present. – 2012-02-27
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0TeX tip: unless one *really* means subtraction, one writes `$\varepsilon$-$\delta$` or, even better, `$\varepsilon$--$\delta$`, not `$\varepsilon-\delta$` ($\varepsilon$-$\delta$ and $\varepsilon$--$\delta$, not $\varepsilon-\delta$ —the second one's dash is not being rendered here) – 2012-02-27
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0I personally hate it that I didn't learn continuity in terms of topology. I'm personally trying to study basics of Topology: Metric Spaces, Topological Spaces, Connectedness and Companctess. I find the theory, though harder, much more interesting than the bare $\epsilon$-$\delta$ definition. – 2012-04-30
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0I agree that the topological definition should come first. In support of this approach, I have written “A Short Intro to General Topology”, and put it into the Public Domain. Here is the link where you can view it or download it: http://public-domain-materials.weebly.com/a-short-intro-to-general-topology.html. Definition 290 is the relevant item. – 2012-05-18
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1“If you want to build a ship, don’t drum up the men to gather wood, divide the work, and give orders. Instead, teach them to yearn for the vast and endless sea.” -- Antoine de Saint-Exupéry – 2015-10-11