One can easily find a set of functions that are comparable with respect to the big O notation that is, $$f \leq g \Leftrightarrow \exists C\in\mathbb{R}: c \lim_{x\rightarrow\infty} exists x_0 \forall x\geq x_0: |f(x)| \leq C|g(x)|,$$ c|g(x)|,$$ for $f,g \in \mathbb{R}\rightarrow \mathbb{R}$ and continuous.
For example, a set containing all products of polynomials, logarithms, exponential functions and factorial is good. However, it is not a maximal one (e.g. $\ln \ln x$ is lesser than all of its elements).
According to Kuratowski-Zorn lemma, there is a maximal set of comparable functions. My questions are:
- Is there one canonical maximal set of comparable functions with respect to big O notation?
- Is there an explicit construction of any such set?
Edit:
- There was $\lim_{x\rightarrow\infty} |f(x)| \leq C|g(x)|$ which was in my intention an informal notation for the above ($\lim$ about the whole inequality, not only the left hand site).
- However, I do not cling to the definition (especially if some tinkering is going to give more interesting results).
