How to prove $\Theta(g(n))\cup o(g(n))\ne O(g(n))$ How to prove $\Theta(g(n))\cup o(g(n))\ne O(g(n))$ ?

Is there a simple example for understanding? Seems there's a gap between $O(g(n))- \Theta(g(n))$ and $o(g(n))$ just from the definition. But I cannot understand What kind of function lays there?
Consider the function
$$
f(n) = \begin{cases} g(n) &amp; \text{if $n$ is even}, \\ g(n)/n &amp; \text{if $n$ is odd}. \end{cases}
$$