Assume that $f:[0,2\pi]\to [0,2\pi]$ is a continuous function such that $f(0)=f(2\pi)$ and define the function $$g(s)=\int_{\pi}^\pi \frac{\sin f(t+s)\sin f(s)}{\sin t/2} dt.$$ Is $g$ continuous or bounded? Probably not. It is related to Marcel Riesz's famous theorem.

You are right, it is not true. Take $ f(x)=1/\log x $ for $ 0 < x < 1/2 $, then extend to an odd function on $ (1/2,1/2) $, set $f(0)=0$ and extend to the whole interval continuously. Then $g(0)=+\infty$, so $g$ can be neither continuous nor bounded. 

