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 sequenc