The Unapologetic Mathematician

Mathematics for the interested outsider

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 f changes due to its behavior within every neighborhood of a point, no matter how small. If the function has a jump discontinuity at x, for instance, it shows up as an oscillation in \omega_f(x). We now want to investigate to what extent such localized oscillations contribute to the oscillation of f over a spread-out neighborhood of a point.

To this end, let f:X\rightarrow\mathbb{R} be some bounded function on a compact region X\in\mathbb{R}^n. Given an \epsilon>0, assume that \omega_f(x)<\epsilon for every point x\in X. Then there exists a \delta>0 so that for every closed neighborhood N we have \Omega_f(N)<\epsilon whenever the diameter of d(N) is less than \delta. The diameter, incidentally, is defined by

\displaystyle d(N)=\sup\limits_{x,y\in N}\left\{d(x,y)\right\}

in a metric space with distance function d. That is, it’s the supremum of the distance between any two points in N.

Anyhow, for each x we have some metric ball N_x=N(x;\delta_x) so that

\displaystyle\Omega_f(N_x\cap X)<\omega_f(x)+\left(\epsilon-\omega_f(x)\right)=\epsilon

because by picking a small enough neighborhood of x we can bring the oscillation over the neighborhood within any positive distance of \omega_f(x). This is where we use the assumption that \epsilon-\omega_f(x)>0.

The collection of all the half-size balls N\left(x;\frac{\delta_x}{2}\right) forms an open cover of X. Thus, since X is compact, we have a finite subcover. That is, the half-size balls at some finite collection of points x_i still covers X. We let \delta be the smallest of these radii \frac{\delta_{x_i}}{2}.

Now if N is some closed neighborhood with diameter less than \delta, it will be partially covered by at least one of these half-size balls, say N\left(x_p;\frac{\delta_{x_p}}{2}\right). The corresponding full-size ball N_{x_p} then fully covers N. Further, we chose this ball so that the \Omega_f(N_x\cap X)<\epsilon, and so we have

\displaystyle\Omega_f(N)\leq\Omega(N_x\cap X)<\epsilon

and we’re done.

December 8, 2009 Posted by | Analysis, Calculus | 7 Comments