Oscillation in a function is sort of a local and non-directional version of variation. If is a bounded function on some region , and if is a nonempty subset of , then we define the oscillation of on by the formula
measuring the greatest difference in values of on .
We also want a version that’s localized to a single point . To do this, we first note that the collection of all subsets of which contain form a poset as usual by inclusion. But we want to reverse this order and say that if and only if .
Now for any two subsets and , their intersection is another such subset containing . And since it’s contained in both and , it’s above both of them in our partial order, which makes this poset a directed set, and the oscillation of is a net.
In fact, it’s easy to see that if then , so this net is monotonically decreasing as the subset gets smaller and smaller. Further, we can see that , since if we can always consider the difference , the supremum must be at least this big.
Anyhow, now we know that the net has a limit, and we define
where is a subset of containing , and we take the limit as gets smaller and smaller.
In fact, this is slightly overdoing it. Our domain is a topological subspace of , and is thus a metric space. If we want we can just work with metric balls and define
where is the ball of radius around . These definitions are exactly equivalent in metric spaces, but the net definition works in more general topological spaces, and it’s extremely useful in its own right later, so it’s worth thinking about now.
Oscillation provides a nice way to restate our condition for continuity, and it works either using the metric space definition or the neighborhood definition of continuity. I’ll work it out in the latter case for generality, but it’s worth writing out the parallel proof for the - definition.
Our assertion is that is continuous at a point if and only if . If is continuous, then for every there is some neighborhood of so that for all . Then we can check that
for all and in , and so . Further, any smaller neighborhood of will also satisfy this inequality, so the net is eventually within of . Since this holds for any , we find that the net has limit .
Conversely, let’s assume that the oscillation of at is zero. That is, for any we have some neighborhood of so that , and the same will automatically hold for smaller neighborhoods. This tells us that for all , and also . Together, these tell us that , and so is continuous at .