EDB — 13T

[13T]Topics:oscillation.

Given any \(f:X→ℝ\), we define oscillation function \({\operatorname {osc}}(f)\)

1. Note that \({\operatorname {osc}}(f)≥ 0\), and that \(f\) is continuous in \(x\) if and only if \({\operatorname {osc}}(f)(x)=0\).

2. Show that \({\operatorname {osc}}(f)\) is upper semicontinuous.

3. If \((X,d)\) is a metric space, note that

   \[ {\operatorname {osc}}(f) (x) {\stackrel{.}{=}}\lim _{\varepsilon → 0+} \sup \{ |f(y) - f(z)| ~ ,~ d(x,y){\lt}\varepsilon ,d(x,z){\lt}\varepsilon \} \quad . \]