Are functions in O(n) that are nor in o(n) all in Θ(n)?
Tell the lecturer they're wrong. Take the function
$$ f(n) = \begin{cases} n &amp; n \text{ is even}, \\ 1 &amp; n \text { is odd}. \end{cases} $$
This function is $O(n)$ but neither $o(n)$ nor $\Theta(n)$.

Here is a monotone example, which might be more convincing:
$$ g(n) = \exp \exp \lfloor \ln \ln n \rfloor. $$