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?
Best Answer
You've identified one of the biggest issues already:
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.