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This question has been inspired by a question asked on Mathoverflow about "effectiveness of (epsilon, delta) definition". Most mathematicians have a strong opinion about the need of such definitions: They are a must for mathematics as a discipline and accordingly, they are a must for whom studies mathematics. But, It happens (like the current term) that I do teach calculus to physics students where I cannot come up with a straightforward decision about the use of such definitions. The question is: To what extent are they also a "must" for a person who studies physics?

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Not at all. I think I've never used that formalism in the context of physics; proofs in the mathematics sense aren't all too popular in most branches. –  Danu Feb 23 at 13:26
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I agree with the math people -- those definitions are essential to what you mean by a limit and by continuity, as well as foundational for the whole field of topology, which is critical for several higher-level physics diciplines. Discounting them is silly. Difficulty level is irrelevant. –  Jerry Schirmer Feb 23 at 13:45
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Frankly, this definition isn't really all that hard if when saying it you also make a drawing, denoting $\varepsilon$, $\delta$ and the neighborhood of limit point. (I realized this when I first studied analysis at university and tried to understand all this epsilon-delta — it appeared really simple as compared to non-rigorous handwaving). –  Ruslan Feb 23 at 14:27
    
Possible duplicate: physics.stackexchange.com/q/234/2451 –  Qmechanic Feb 23 at 15:04
    
(IMO this is not a duplicate question.) At the very least epsilon and delta give some flavor of rigor, and that's valuable, even to engineers and experimentalists. But it's not essential for that audience. I think that if there's any chance that a student will make a career in physics, then yes, include it. Apocryphal story-- Student: How much math does a physics student need to know? Victor Weiskopf: More. –  garyp Feb 23 at 17:11

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Physics needs math, and math needs the definition.

I learned calculus without the epsilon-delta definition of a limit, then learned the epsilon-delta definition. I think it makes sense to teach it that way -- understand the idea of a limit, and see some useful applications of it, then learn a rigorous definition. So my answer is: it's a "must" for physics, to be presented at the right time.

Ruslan's comment above said it well: "Frankly, this definition isn't really all that hard if when saying it you also make a drawing, denoting ε, δ and the neighborhood of limit point. (I realized this when I first studied analysis at university and tried to understand all this epsilon-delta — it appeared really simple as compared to non-rigorous handwaving)." It just doesn't make sense to go through life without that clear understanding of what a limit really is.

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