Exercise 5, § 1

Let $f$ be a vector function defined on a compact interval $I = [a, b] \subset \mathbf{R}$, taking its values in a complete normed space $E$; one says that $f$ is of bounded variation on $I$ if there exists a number $m > 0$ such that, for every finite strictly increasing sequence $(x_i)_{0 \leq i \leq n}$ of points of $I$ such that $x_0 = a$ and $x_n = b$, one has

a) Show that $f(I)$ is relatively compact in $E$ (argue by contradiction).

b) Show that $f$ is regulated on $I$ (prove that when $x$ tends to a point $x_0 \in I$, remaining $> x_0$, the function $f$ cannot have two different cluster points, and use $a$).