When can we assuredly say that a function is little o of some other function?
Your example doesn't work, since in fact $\log x = o(x)$. However, the function $\phi$ below is $O(x)$ but not $o(x)$ or $\Omega(x)$:
$$
\phi(x) = \begin{cases} x &amp; \text{if $\lfloor x \rfloor$ is even}, \\ 1 &amp; \text{if $\lfloor x \rfloor$ is odd}. \end{cases}
$$