Big-O Notation Statement True?

Question

Considering functions f and g, is the following true?

$ (f \in O(g)) \implies (f \in \Theta(g)) \lor (f \in o(g))$

If not, can you please state an example? Despite thinking hard, i could not find one.

Thanks a lot!

Answer 1

No, this is not true. For example, let $f$ be the constant $1$ function, and define $g$ by $g(2n) = 1/n$ and $g(2n+1) = 1$.