Limits of Functions
Okay, we know what it is for a net to have a limit, and then we used that to define continuity in terms of nets. Continuity just says that the function’s value is exactly what it takes to preserve convergence of nets.
But what if we have a bunch of nets and no function value? Like, if there’s a hole in our domain — as there is at 
for the function 
— we certainly shouldn’t penalize this function just on a technicality of how we presented it. Well there may be a hole in the domain, but we still have sequences in the domain that converge to where that hole is. So let’s take a domain 
, a function 
, and a point 
. In particular, we’re interested in what happens when 
is in the closure of 
, but not in itself.
Now we look at all sequences 
which converge to . There’s at least one of them because , but there may be quite a few. Each one of these sequences has an opinion on what the value of 
should be at . If they all agree, then we can define the limit of the function 
where 
is any one of these sequences. In the case of we see that at every point other than our function takes the value 
. Thus on any sequence converging to (but never taking 
) the function gives the constant sequence . Since they all agree, we can define the limit 
.
If a function has a limit at a hole in its domain, we can use that limit to patch up the hole. That is, if our point is in the closure of but not in itself, and if our function has a limit at , then we can extend our function to 
by setting 
. Just like we by default set the domain of a function to be wherever it makes sense, we will just assume that the domain has been extended to whatever boundary points the function takes a limit at.
On the other hand, we can also describe limits in terms of neighborhoods instead of sequences. Here we end up with formulas that look like those we saw when we defined continuity in metric spaces. A function has a limit 
at the point if for every 
there is a 
so that 
implies 
. Going back and forth from this definition to the one in terms of sequences behaves just the same as going back and forth between net and neighborhood definitions of continuity.
To a certain extent we’re starting to see a little more clearly the distinct feels of the two different approaches. Using nets tells us about approaching a point in various systematic ways, and having a limit at a point tells us that we can understand the function at that point by understanding any system along which we can approach it. We can even replace the limiting point by the convergent net and say that the net is the point, as we did when first defining the real numbers. Using neighborhoods, on the other hand, feels more like giving error tolerances. A limit is the value the function is trying to get to, and if we’re willing to live with being wrong by 
, there’s a way to pick a 
for how wrong our input can be and still come at least that close to the target.
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December 19, 2007 -
Posted by John Armstrong |
Analysis, Calculus
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Why talk about nets at all in a metric space? You can get along with sequences. Nets are only needed in more complicated topoligical spaces.
My feeling towards sequences is also split. On hand hand, they are easy to define and imagine, since they are countable. On the other hand, the sets of all sequences is mighty big, and thus the sequence definition of continuity is not really constructive.
In fact, going back and forth between the two definitions of continuity requires the axiom of choice (on countable sets). So epsilon-delta looks more constructive.
Of course, a single sequence (like the power series for exp) is constructive too.
First off, I’m not a constructivist, so considerations like that aren’t really very persuasive for me.
Secondly, notice that I do immediately move to talking about sequences. But I keep nets around because there are some concepts that naturally give rise to nets, not sequences. For example, the limit involved in Riemann integration is most clearly expressed as that of a net over the directed set of marked partitions, rather than that ugly limit over “mesh size” many authors keep using.