Remark 1

When $f$ is continuous prop. 5 can be proved more simply: since $f$ is uniformly continuous on $[x_0,\ x]$ there exists a $\rho > 0$ such that on every interval of length $\leq \rho$ contained in $[x_0,\ x]$ the oscillation of $f$ is $\leq \frac{\varepsilon}{x - x_0}$; for every subdivision of $[x_0,\ x]$ into intervals $[x_i,\ x_{i+1}]$ of length $\leq \rho$ and every choice of $t_i$ in $[x_i,\ x_{i+1}]$ for $0 \leq i \leq n-1$ the step function $f_1$ equal to $f(t_i)$ on $[x_i,\ x_{i+1}]$ ($0 \leq i \leq n-1$), and to $f(x)$ at the point x, is such that $\| f(y) - f_1(y) \| \leq \frac{\varepsilon}{x - x_0}$ on $[x_0, x]$; if $g_1$ is a primitive of $f_1$ one has

so the relation (1) follows immediately from the mean value theorem.

In the rest of this chapter we shall confine ourselves to the study of primitives of regulated functions on an interval I. For such a function $f$, with values in E, a primitive $g$ of $f$, and for two arbitrary points $x_0, x$ of I, the element $g(x) - g(x_0)$ of E (which is clearly the same, no matter which primitive $g$ of $f$ one considers) is called the integral of the function $f$ from $x_0$ to $x$ (or over the compact interval $[x_0, x]$) and is denoted by $\int_{x_0}^x f(t) dt$ or $\int_{x_0}^x f$. This name and notation have their origin in prop. 5 of II, p. 57, which shows that an integral can be approximated arbitrarily closely by a Riemann sum; more particularly, one can write, taking subdivisions of $[x_0, x]$ into equal intervals,

In other words, the element $\frac{1}{x - x_0} \int_{x_0}^x f(t) dt$ is the limit of the arithmetic means of the values of $f$ at the left-hand endpoints of the intervals of a subdivision of $[x_0, x]$ into equal intervals; one also calls it the mean (or mean value) of the function $f$ on the interval $[x_0, x]$.

By definition, the function $x \mapsto \int_{x_0}^x f(t) dt$ is none other than the primitive of $f$ which vanishes at the point $x_0 \in I$; one also denotes it by $\int_{x_0}^x f(t) dt$ or $\int_{x_0}^x f$.