A problem about master theorem and recursion
Taking $n=2^k$, we have
$$T(n)=2T\left(\frac{n}{2}\right)+f(n)=\\=2\left[2T\left(\frac{n}{2^2}\right) +f(n)\right]+f(n)=2^2T\left(\frac{n}{2^2}\right)+2f(n)+f(n)=\\=2^3T\left(\frac{n}{2^3}\right)+2^2f(n)+2f(n)+f(n)=\cdots=2^kT(1)+f(n)[2^{k-1}+\cdots+1]=\\=2^kT(1)+f(n)(2^k-1) = nT(1)+f(n)(n-1)$$
Now we have 2 cases: 1. $T,f$ are defined only for $2$s powers and 2. $T,f$ are defined on $\mathbb{N}$.

Considering non negative functions, condition $T \in O(f)$    gives
$$ f(n)(n-1) \leqslant T(n) \leqslant C f(n)$$
which is impossible for any below boundedness for $f$.

If we demand $T$ defined on $\mathbb{N}$, then we come to necessity to define $T\left(\frac{n}{2}\right)$ which can be filled by OP.