From Local Oscillation to Neighborhoods
When we defined oscillation, we took a limit to find the oscillation “at a point”. This is how much the function changes due to its behavior within every neighborhood of a point, no matter how small. If the function has a jump discontinuity at , for instance, it shows up as an oscillation in . We now want to investigate to what extent such localized oscillations contribute to the oscillation of over a spread-out neighborhood of a point.
To this end, let be some bounded function on a compact region . Given an , assume that for every point . Then there exists a so that for every closed neighborhood we have whenever the diameter of is less than . The diameter, incidentally, is defined by
in a metric space with distance function . That is, it’s the supremum of the distance between any two points in .
Anyhow, for each we have some metric ball so that
because by picking a small enough neighborhood of we can bring the oscillation over the neighborhood within any positive distance of . This is where we use the assumption that .
The collection of all the half-size balls forms an open cover of . Thus, since is compact, we have a finite subcover. That is, the half-size balls at some finite collection of points still covers . We let be the smallest of these radii .
Now if is some closed neighborhood with diameter less than , it will be partially covered by at least one of these half-size balls, say . The corresponding full-size ball then fully covers . Further, we chose this ball so that the , and so we have
and we’re done.