Exercise 2, § 2

a) Let $\mathbf{f}$ be a regulated function on an interval $[0, a]$ such that the integral $\int_0^a \frac{\mathbf{f}(t)}{t}\, dt$ is convergent. Show that the integral $g(x) = \int_0^x \mathbf{f}(t)\, dt$ is convergent and that $g$ admits a right derivative at the point $x = 0$ (integrate suitably by parts).

b) Give an example of a real function $f$ such that the integral $g(x) = \int_0^x f(t)\, dt$ is convergent and admits a right derivative at the point 0, and yet $f(t)/t$ does not have an integral on ]0, a] (take for $f(x)/x$ the derivative of a function of the form $\cos \varphi(x)$, where $\varphi$ tends to $+\infty$ as $x$ tends to 0).