Definition 1

Given a vector function $f$ defined on an interval $I \subset \mathbf{R}$ we say that a function $g$ defined on $I$ is a primitive of $f$ if $g$ is continuous on $I$ and has a derivative equal to $f(x)$ at every point $x$ of the complement (with respect to $I$) of a countable subset of $I$.

If also $g$ admits a derivative equal to $f(x)$ at every point $x$ of $I$, one says that $g$ is a strict primitive of $f$.

With this definition, one sees that the real function $f$ considered above admits a primitive equal to $|x|$.

It is clear that if $f$ admits a primitive on $I$ then every primitive of $f$ is also a primitive of every function which is equal to $f$ except at the points of a countable subset of $I$. By an abuse of language one speaks of a primitive on $I$ of a function $f_0$ defined only on the complement (with respect to $I$) of a countable subset of $I$: this will be the primitive of every function $f$ defined on $I$ and equal to $f_0$ at the points where $f_0$ is defined.