Exercise 6, § 1

For a function $f$, with values in a complete normed space $E$, and defined on an open interval $I \subset \mathbf{R}$, to be equal to a regulated function on $I$ at all points of the complement of a countable subset of $I$, it is necessary and sufficient that it satisfy the following condition: for every $x \in I$ and every $\varepsilon > 0$ there exist a number $h > 0$ and two elements $a,\ b$ of $E$ such that one has $\|f(y) - a\| \leq \varepsilon$ for every $y \in [x, x + h]$, except for at most a countably infinite number of points of this interval, and $\|f(z) - b\| \leq \varepsilon$ for all $z \in [x - h, x]$ except for at most a countably infinite number of points of this interval.