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.
December 8, 2009
Posted by John Armstrong |
Analysis, Calculus |
7 Comments
