Put $\varphi(t)= 1- \cos \;t\;\;\;$ if $\;\;\;0 \leq t \leq 2 \pi$, $\varphi(t) = 0$ for all other real $t$. For $-\infty < x < \infty $, define
$$ f(x)= 1,\;\;\;\;\;\;\;\;\;\;g(x) = \varphi'(x), \;\;\;\;\;\;\;\;\; h(x)\int^x_{-\infty} \,\varphi(t) dt. $$
Verify the following statements about convolutions of these functions:
$(i)$ $(f\ast g)(x) = 0$ for all x.
$(ii)$ $(g\ast h)(x) = (\varphi \ast \varphi)(x) > 0$ on $(0, 4 \pi)$.
$(iii)$ Therefore $(f\ast g)\ast h = 0$, whereas $f\ast(g\ast h)$ is a postive constant. But convolutions is supposedly associative, by Fubini's theorem. What went wrong?
I do the item $(i)$ and I have trouble with the item $(ii)$ with integrals. This exercise is of Rudin Real and Complex Analysis exercise 15 chapter 8.