I came with this question while trying to get a counterexample for the statement:
if $f$ and $g$ are uniformly continuous functions, $g$ is bounded and $f$ is not necessarily bounded then $fg$ is uniformly continuous.
That is a common question in analysis books and is, indeed, already answered in this forum. But the answers, both in books and in the forum, are always the same( or slight variations of it):
$f(x) = x$ and $g(x) = sin\,x$
So, are there any other counterexamples? In others terms, can we rephrase the statement imposing that $g$ "doesn't look like" $sin\,x$ to make it true?
An example, that I have no evidence of being true, would be:
if $f$ and $g$ are uniformly continuous functions, $g$ is bounded and not periodic, $f$ is not necessarily bounded. Then $fg$ is uniformly continuous.