Comparing $\mathcal{O}$ and $\Omega$

Question

By the definition of Big $\mathcal{O}$ and $\Omega$ of any function $f$, is it correct to say that the only common element in the two sets is $f$ and the two sets are otherwise disjoint?

Answer 1

No, Consider $f(x) = c x^2$ so that $f \in \Omega(x^2)$ and $f \in \mathcal{O}(x^2)$.

Now, you can see that the intersection contains infinitely many functions.