Division by 2 on hyperelliptic curves and jacobians

Let $K$ be an algebraically closed field of characteristic different from 2, $g$ a positive integer, $f(x)$ a degree $(2g+1)$ polynomial with coefficients in $K$ and without multiple roots, $C: y^2=f(x)$ the corresponding genus $g$ hyperelliptic curve over $K$ and $J$ the jacobian of $C$. We identify $C$ with the image of its canonical embedding into $J$ (the infinite point of $C$ goes to the zero point of $J$). For each point $P=(a,b)\in C(K)$ there are $2^{2g}$ points $\frac{1}{2}P \in J(K)$. We describe explicitly the Mumford represesentations of all $\frac{1}{2}P$. The rationality questions for $\frac{1}{2}P$ are also discussed.