As we’ve seen, when our target is a higher-dimensional real space continuity is the same as continuity in each component. But what about when the source is such a space? It turns out that it’s not quite so simple.
One thing, at least, is unchanged. We can still say that is continuous at a point if . That is, if we have a sequence of points in (we only need to consider sequences because metric spaces are sequential) that converges to , then the image of this sequence converges to .
The problem is that limits themselves in higher-dimensional real spaces become a little hairy. In there’s really only two directions along which a sequence can converge to a given point. If we have a sequence converging from the right and another sequence converging from the left, that basically is enough to establish what the limit of the function is (and if it has one). In higher-dimensional spaces — even just in — we have so many possible approaches to any given point that in order to avoid an infinite amount of work we have to use something like the formal definition of limits in terms of metric balls. That is
The function has limit at the point if for every there is a so that implies .
We just consider the case with target since higher-dimensional targets are just like multiple copies of this same definition, just as we saw for continuity.
Now, let’s look at a few examples of limits to get an idea for why it’s not so simple. In each case, we will be considering a function which is bounded near (since just blowing up to infinity would be too easy to be really pathological) and even with nice limits along certain specified approaches, but which still fail to have a limit at the origin.
First off, let’s consider . If we consider approaching along the -axis with the sequence or we find a limit of . However, if we approach along the -axis with the sequence or we instead find a limit of . Thus no limit exists for the function.
Next let’s try . Now the approaches along either axis above all give the limit , so the limit of the function is , right? Wrong! This time if we approach along the diagonal with the sequence we get the limit . So we have to consider directions other than the coordinate axes.
What about ? Approaching along the coordinate axes we get a limit of . Approaching along any diagonal with the sequence the calculations are a bit hairier but we still find a limit of . So approaching from any direction we get the same limit, making the limit of the function , right? Wrong again! Now if we approach along the parabola with the sequence we find a limit of , and so the limit still doesn’t exist. By this point it should be clear that if straight lines aren’t enough to simplify things then there are just far too many curves to consider, and we need some other method to establish a limit, which is where the metric ball definition comes in.
Now I want to go off on a little bit of a rant here. It’s become fashionable to not teach the metric ball definition — — proofs, as they’re often called — at the first semester calculus level. It’s not even on the Calculus AB exam. I’m not sure when this happened because I was taught them first thing when I took calculus, and it wasn’t that long between then and my first experience teaching calculus. But it’d have to have been sometime in the mid-’90s. Anyway, they don’t even teach it in most college courses anymore. And for the purposes of calculus that’s okay, since as I mentioned above you can easily get away without them when dealing with single-variable functions. They can even survive the analogues of — proofs that come up when dealing with convergent sequences in second-semester calculus.
The problem comes when students get to third semester calculus and multivariable functions. Now, as we’ve just seen, there’s no sure way of establishing a limit. We can in some cases establish the continuity of simple functions (like coordinate projections) and then use limit laws to build up a larger class. But this approach fails for functions superficially similar to the pathological functions listed above, but which do have limits which can be established by an — proof. We can establish that certain limits do not exist by techniques similar to those above, but this requires some ingenuity in choosing two appropriate paths which give different results. There are one or two other methods that work in special cases, but nothing works like an — proof.
But now we can’t teach — proofs to these students! The method is rather more complicated when we’ve got more than one variable to work with, not least because of the more complicated distance formula to work with. What used to happen was that students would have developed some facility with — proofs back in first and second semester calculus, which could then be brought to bear on this new situation. But now they have no background and cannot, in general, absorb both the logical details of challenge-response — proofs and the complications of multiple variables at the same time. And so we show them a few jury-rigged tricks and assure them that within the rest of the course they won’t have to worry about it. I’d almost rather dispense with limits entirely than present this Frankenstein’s monstrosity.
And yet, I see no sign that the tide will ever turn back. The only hope is that the movement to make statistics the capstone high-school course will gain momentum. If we can finally wrest first-semester calculus from the hands of the public school system and put all calculus students at a given college through the same three-semester track, then the more intellectually rigorous institutions might have the integrity to put proper limits back into the hands of their first semester students and not have to worry about incoming freshmen with high AP scores covering for shoddy backgrounds.