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While studying single and multivariable calculus during my first year some people complained that calculus wasn't rigorous enough, when I asked about this no one seemed to be able to really specify exactly what was not rigorous about it. So I want to ask if this is true or if my friends tried to be smarty pants? My professors mentioned nothing about this.

The only such thing I can think of is that we considered $\mathbb{R}$ (and $\mathbb{R}^n$) to be given and just kind of "the number line of every number you possibly can think of". We didn't care about the construction of the reals at all. But I'm pretty sure that this is not what they meant.

I don't think I will take any sturdy course in real analysis so I want to ask if the standard definitions and proofs involving limits, derivatives, differentiability, continuity, integrals etc one stumbles upon in calculus is somehow "simplified" in calculus and made more formal and "clear" in later and more advanced courses in real analysis?

If this is true, does it exist any good examples which can illustrate this for someone who is slightly afraid of epsilon and deltas?

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It depends quite a bit on your textbook! Were you taught the epsilon-delta definition of a limit? If not, that's a source of non-rigor. – Grumpy Parsnip Jul 24 '14 at 6:03
Did you "manipulate" differentials in your course? That's non-rigorous because infinitesimals aren't real numbers. – user_of_math Jul 24 '14 at 6:16
@GrumpyParsnip We mostly used Calculus: a Complete Course by Robert A. Adams and we were actually forced to prove some limits using the epsilon-delta definition in one of our exams if I remember it correctly. – John Smith Jul 24 '14 at 6:16
@user_of_math Ah, yes that we did! We kind of treated treated them like reals without any explanation and (I think) without any formal definition. Thanks! – John Smith Jul 24 '14 at 6:20
Just to put this out there: Non-standard analysis does deal with infinitesimals in such a way that justifies "manipulating" them like numbers (subject to certain caveats). Also, if you think of differentials not as small numbers, but as the shorthand for the limit of an approximation then they behave a lot like numbers in many contexts. – user76844 Jul 24 '14 at 12:59
up vote 8 down vote accepted

You've identified one of the biggest issues already:

we considered $\Bbb R$ (and $\Bbb R^n$) to be given and just kind of "the number line of every number you possibly can think of"

As a result, there are some very important theorems that I'll bet you didn't prove formally (although you probably did draw pictures corresponding to them), such as the Intermediate Value Theorem. Ultimately, the IVT is a topological statement about how the reals behave that, in a way, formalizes what we mean by "every number you possibly can think of," at least in terms of filling in holes and making $\mathbb{R}$ complete. Not discussing what a complete metric space actually is means that there's going to be quite a bit missing.

Likewise, the Extreme Value Theorem, which is the key step in proving the Mean Value Theorem, is something that one usually doesn't approach without some basic topological knowledge (or a significantly more in-depth knowledge of sequences than is typical for a general calculus course). Both of these two theorems I've mentioned really rely on that concept of "completeness," or "not having any holes."

But otherwise, the course is probably pretty rigorous - the $\epsilon-\delta$ approach isn't lacking of anything from a technical view, and there's quite a bit you can do only using it.

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'Another classical result that fails to hold constructively in its usual form is the well-known intermediate value theorem... However, it can be shown that, constructively, the [IVT] is "almost" true in the sense that... This example illustrates how a single classical theorem "refracts" into several constructive theorems.' The Continuous and the Infinitesimal p276, John L Bell. – mistermarko Jul 24 '14 at 9:23
@mistermarko: What are you bringing in constructivism for? – user2357112 Jul 24 '14 at 10:40
Because it leads to intuitionism with the result that 'In 1980 Richard Vesley showed that a natural notion of infinitesimal can be developed within intuitionistic mathematics. His idea was that an infinitesimal should be a "very small" real number in the sense of not being known to be distinguishable - that is, strictly greater than or less than - zero.' ibid p281. Seems relevant. – mistermarko Jul 24 '14 at 11:03
@mistermarko Nothing you have said (in spite of your accusations of "orthodoxy") has indicated why you consider some version of non-standard analysis to be the best way to do calculus; and it certainly seems entirely unrelated to the original question about whether the calculus taught in an early course is rigorous or not. Regardless of the validity of your viewpoint, your comments and answer seem to be entirely about preaching a philosophical viewpoint. – user61527 Jul 25 '14 at 7:05
@mistermarko I have no idea what you mean by "a real number is never exact anyway." By real number, I mean a mathematically precise object, namely an element of the field $\mathbb{R}$; it has nothing to do with measurement errors or physical interpretations. One can make non-standard analysis perfectly rigorous, but not by defining an infinitesimal in the imprecise way that you seem to be defining it. – user61527 Jul 25 '14 at 7:48

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