Example

If the interval I is bounded, and if $f$ is bounded on I, then the integral $\int_I f(t)\,dt$ always exists, for, by the mean value theorem, one has, for $y \leq \alpha \leq \beta \leq z$,

and it suffices to take $\alpha - a$ and $b - \beta$ small enough for the Cauchy criterion be satisfied.

One may note that in this case a primitive of $f$ on I does not necessarily have a right (resp. left) derivative at the left-hand endpoint (resp. right-hand endpoint) of I (when this number is finite) contrary to the situation when I is compact and $f$ is regulated on I (cf. II, p. 33, exerc. 1).