According to Oppenheim's signal and systems solution book and other solutions like this
$$ Ev(\cos(4\pi t)u(t)) = 1/2 \times ( \cos(4\pi t)u(t) + \cos(4\pi t)u(-t)) = 1/2(\cos(4\pi t)) $$
$$ \textrm{for }-\infty < t < \infty $$
is periodic and its period $1/2$.
But I think it is not equal $ 1/2(\cos 4\pi t) $ at $ t = 0 $ so it is not periodic because at $ t = 0 $ it is equal to $ 1 $ different from other multiples of period.
I have also curious about the value of $ u(t)+u(-t) $. Is it $0$, $1$, or $2$ ?