There are $C^{\infty}$ test functions in $L^{1}(0, \infty)$ that make the integral value of $\int_{0}^{\infty}(\sin x/x) \phi(x) dx$ range from 0 to $\infty$. What does narrowing these test functions down so that the integral falls in the interval $(\pi/2 - \varepsilon, \pi/2 + \varepsilon)$ tell us distribution theorywise?

What is the transformation $T: R^2 \rightarrow R^2$ that sends the area bounded by the curve sinx/x for the given interval to say half the upper unit disk.

EDIT: These test functions awfully appear like the Fourier sine integrals with an extra division by x. And this division causes a good deal of the "Fourier" transforms to get thrown out of Ln(0,∞). As for critically bounded supersymmetry, extra conditions are most likely needed.

Incidentally, Eli Maor in "Trigonometry Delight" chap. 10 sin x/x gives references for proof of the integral value of sinx/x in Erwin Kreyszig's adv. math. eng. '79 pp.735-736 and in Courant's "diff. and int. calculus". Kreyszig's latest editions are expanded enough that frankly I couldn't find it. It's nothing but a simple residue problem alright. Also see comments, and here. Where are the mathoverflow jokes?