Remark 4

The approximation of an integral by Riemann sums is closely connected to one of the historical origins of the concept of an integral, the problem of the measure of areas. We shall come back to this point in the Book on Integration which is devoted to generalizations of the concept of integral to which this problem leads; in these generalizations the functions to be "*integrated*" are not necessarily defined on a subset of $\mathbf{R}$; moreover, even when one deals with (not necessarily regulated) real functions $f$ of a real variable for which one can define an integral $\int_{x_0}^x f(t)\,dt$, the function $x \mapsto \int_{x_0}^x f(t)\,dt$ is not always a primitive of $f$, and there exist functions which have a primitive but are not "integrable" in the sense to which we allude.