Why is $\frac{\phi(x)-\phi(-x)}{x}$ for smooth $\phi$ bounded at $x=0$? If i set $\phi(x) = \sqrt{|x|}$, it definitely not bounded. I saw this on page 293 of
http://www.math.ucdavis.edu/~hunter/book/pdfbook.html
where in example it 11.7 it is claimed the integrand (which is an expression like mine) is bounded?