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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.

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So: $f$ is uniformly continuous. –  Gerald Edgar Feb 13 '13 at 23:03
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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.

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