Let \begin{align} f \in C^1 \text{ with } f(0) , f(\infty) \in \Bbb R \text{ and } b>a>0 \tag{1}. \end{align} Show \begin{align*} \int_0^\infty \frac{f(ax)-f(bx)}{x}dx = \left( f(0) - f(\infty) \right) \log\left( \frac{b}{a} \right). \end{align*}
There is a proof of this identity in this answer.
How do we justify the application of Fubini theorem to interchange the order of integration.
I don't see why (1) is sufficient to guarantee the finiteness of
\begin{align}
\int_0^\infty \int_a^b \left| f'(yx) \right| dy dx ,
\end{align}
and
\begin{align}
\int_a^b \int_0^\infty \left| f'(yx) \right| dx dy .
\end{align}
