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We consider integrals of the form \[ I_f(a, b)=∫_0^∞ \frac{f(a x)-f(b x)}{x} d x, \] where $f$ is a continuous function (real or complex) on $(0,∞)$ and $a,b>0$. If $f(x)$ tends to a non-zero limit at 0 , then the separate integrals of $f(ax)/x$ and $f(bx) / x$ diverge at 0, and a similar comment applies at infinity. The point is that under suitable conditions, the integral of the difference converges.