EDB — 143

[143]Prerequisites:[13T]↺↻. Let \(I=[a,b]\) be closed and bounded interval. Show that

• \(f:[a,b]→ℝ\) is regulated if and only if

• for any \(\varepsilon {\gt}0\), there exists a finite set of points \(P⊂ I\) such that, for every \(J\subseteq I\) with \(J\) an open interval that does not contain any point of \(P\), the oscillation of \(f\) in \(J\) is less than \(\varepsilon \).