On the absence of exact means for some bounded functions

It is shown that there exist almost periodic functions $f(t)$ such that the function
$$ \varphi(t)=\frac1{c+\int^t_0f(t)\,dt} $$
is bounded but does not have an exact mean value. This fact implies that there are first-order Bernoulli differential equations with almost periodic coefficients such that some bounded solutions do not have exact mean values.
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