Can subtracting o(1) from the parameter of a function change its Θ-class?
Let $g(n) = 1/n$ and define
$$ f(n) = \begin{cases} n &amp; \text{if $n$ is an integer}, \\ \log \lceil n \rceil &amp; \text{otherwise}. \end{cases} $$
For an integer $n&gt;1$, $f(n) = n$ while $f(n-g(n)) = \log n$.