Let $X\subset \mathbb R^m$ a compact subset and $f:X\to \mathbb R$ bounded. We know the oscillation in a set is defined as $\omega(f,X)=\sup \{|f(x)-f(y)|;x,y\in X\}$ and at a point as $\omega(f,x)=\lim_{\delta\to 0}\omega(f,B(x,\delta)\cap X)=\inf_{\delta>0}\omega(f,B(x,\delta)\cap X)$.
I would like to show there is a point $x_0\in X$ such that the oscillation reaches its maximum and not necessarily its minimum. My first thought was to show the oscillation is a continuous function in a compact, but in this case the oscillation must always reach a minimum which is not necessarily true according to the question.