Definition 1

One says that a vector function $f$, piecewise regulated on an interval $[a,\ b[$ of $\overline{\mathbf{R}}$, admits an integral on this interval if $f$ admits a primitive on $[a,\ b[$*; if* $g$ is any one of the primitives of $f$ on $[a,\ b[$*, and* $x_0$ and $x$ are any two points of $[a,\ b[$*, one calls the element $g(x) - g(x_0)$ the integral of $f$ from $x_0$ to $x$, and one denotes it by $\int_{x_0}^x f(t)\,dt$.

This concept clearly agrees with that defined when the interval $[x_0,\ x]$ contains none of the points $c_i$.

The remarks which precede def. 1 show that for $f$ to have an integral on $[a,\ b[$ it is necessary and sufficient that its restriction to each of the intervals $[c_i,\ c_{i+1}[$ should admit an integral over this interval. In other words, one reduces to the case where $f$ is regulated on a non-compact interval $I \subset \mathbf{R}$, with endpoints $a,\ b$ ($a < b$), and where: 1 either one of the numbers $a,\ b$ (at least) is infinite; 2 or $f$ is not regulated on a compact interval containing at least one of the points $a,\ b$ (these two hypotheses not being mutually exclusive). For $f$ to have an integral over $I$ it is necessary and sufficient that the integral $\int_a^x f(t)\,dt$ approaches a limit when the point $(x,\ y)$ approaches $(a,\ b) \in \overline{\mathbf{R}}^2$ while remaining in $I \times I$: and this limit is no other than $\int_a^b f(t)\,dt$ according to def. 1. By an abuse of language, instead of saying that $f$ has an integral over $I$, one says that the integral $\int_a^b f(t)\,dt$ is convergent (or converges).