13.2 Regulated functions[2CT]

(Note in particular that every monotonic function is regulated, and every continuous function is regulated.)

E348

[142] Show that a regulated function \(f:[a,b]→ℝ\) is bounded.

E348

[143]Prerequisites:9. 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 \(\varepsilon \).

E348

[144] Let \(I=[a,b]\). Let \(V\) be the set of functions \(f:[a,b]→ℝ\) that are piecewise constant; it is the vector space generated by \({\mathbb 1}_ J\), all the characteristic functions of all intervals \(J\subseteq I\). Prove that the closure of \(V\) (according to uniform convergence) coincides with the space of regulated functions.

So the space of regulated functions, endowed with the norm \(\| ⋅\| _∞\), is a Banach space.

See also exercises 4, 4, 4 and 5.