Exercise 38, § 1

Let $\mathbf{f}$ be a regulated vector function for $x \geqslant 0$, continuous at the point $x = 0$, and such that the integral $\int_a^{+\infty} \mathbf{f}(x) dx / x$ converges for $a > 0$. Show that for $a > 0$ and $b > 0$ the integral $\int_0^{+\infty} \frac{\mathbf{f}(ax) - \mathbf{f}(bx)}{x} dx$ converges and is equal to $\mathbf{f}(0) \log a / b$.