Prove / disprove: if $f(n)=O(g(n))$ then $2^{f(n)}=O (2^{g(n)})$
This is related to: Big-O: If $f(n)=O(g(n))$, prove $2^{f(n)}=O(2^{(g(n)})$
But I don't see how the answer there is right (I commented on that there) and I have a question about a different approach.
Can we use that since $2^x$ is a continuous positive function then it keeps signs of inequalities? i.e $a<b$ then $2^a<2^b$?