Let $T$ be a function.
Is it true that if $\exists f\forall n,m> 0.\\ \frac m {f(n)} \leq T(n,m)\leq m$
Then $\exists g.T(n,m)=\Theta(m\cdot g(n))$?
In words: is such a case, is there a function $g$ dependent only on $n$ that satisfies the above?
It looks to me true, but I am really stuck and don't know how to prove it rigorously.