Show that the union of Θ and o is not O
An obvious function is 
$$
f(n) = \begin{cases}
n \log n &amp; \text{if $n$ is even}, \\
0 &amp; \text{if $n$ is odd}. 
\end{cases}
$$

It's in $O (n \log n)$, it's not in $o (n \log n)$ and not in $\Theta(n \log n)$.

By the way: Every function in $\Theta(f(n)))$ is in $O(f(n))$, just take the larger constant from the $\Theta$ definition. And every function in $o(f(n))$ is also in $O(f(n))$, choose $c = 1$ and make $n$ large enough. But not the other way around.

$O(f(n))$ contains functions that have infinitely many values between $c_1 f(n)$ and $c_2 f(n)$, but also infinitely many small values. So a function $g(n) = f(n)$ for infinitely many $n$, and $g(n) = 0$ for infinitely many $n$ where $f(n) \ne 0$ is in $O (f(n))$, but not in $\Theta(f(n))$ and not in $o (f(n))$.