Proposition 1

Let $f$ be a vector function defined on $I$ with values in $E$; if $f$ admits a primitive $g$ on $I$ then the set of primitives of $f$ on $I$ is identical to the set of functions $g + a$, where $a$ is a constant function with its values in $E$.

Indeed, it is clear that $g + a$ is a primitive of $f$ for any $a \in E$; on the other hand, if $g_1$ is a primitive of $f$ then $g_1 - g$ admits a derivative equal to 0 except at the points of a countable subset of $I$, and thus is constant (I, p. 17, corollary).

One says that the primitives of a function $f$ (when they exist) are defined “up to an additive constant”. To define a primitive of $f$ unambiguously it is enough to assign it (arbitrarily) a value at a point $x_0 \in I$; in particular, there exists one and only one primitive $g$ of $f$ such that $g(x_0) = 0$; for every primitive $h$ of $f$ one has $g(x) = h(x) - h(x_0)$.