# The Unapologetic Mathematician

## 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