Corollary 2

Let f_i (1 \leq i \leq n) be n regulated functions on an interval I, each f_i having its values in a complete normed space E_i over \mathbf{R} (1 \leq i \leq n). If g is a continuous map of the subspace \prod_{i=1}^{n} \overline{f_i(I)} of \prod_{i=1}^{n} E_i into a complete normed space F over \mathbf{R}, then the composite function x \mapsto g(f_1(x), f_2(x), \ldots, f_n(x)) is regulated on I.

Indeed, it clearly satisfies the conditions of th. 3 of II, p. 54.

Thus one sees that if f is a regulated vector function on I, then the real function x \mapsto \|f(x)\| is also regulated. Further, the real regulated functions on I form a ring;

moreover, if $f$ and $g$ are two real regulated functions, then $\sup(f, g)$ and $\inf(f, g)$ are regulated.