Dyadic Map Derivative at One-Half

Let $f:(0,1) \rightarrow (0,1)$ be a differentiable function with a continuous derivative such that for every positive integer $n$ and odd positive integer $a < 2^{n}$, there exists an odd positive integer $b < 2^{n}$ such that $f\left(\frac{a}{2^{n}}\right) = \frac{b}{2^{n}}$. Determine the set of possible values of $f^{\prime}\left(\frac{1}{2}\right)$.