Proposition 4

Every monotone real function $f$ on an interval $I \subset \mathbf{R}$ is regulated, and every primitive of $f$ is convex on $I$.

Indeed, $f$ satisfies the criterion of th. 3 of Gen. Top., IV, p. 350, prop. 4; the second part of the proposition follows from cor. 1, from II, p. 55, and from prop. 5 of I, p. 27.