Let $C$ be some smooth proper curve of genus $g$ over an algebraically closed field $k$. In order to understand special divisors on C one may consider the following function c(r), which I will call Clifford's function. It assigns to a number $0 \leq r \leq g-1$ the (minimal degree of a divisor such that r(D) = dim|D| = r) - 2r. Clifford's theorem states that $c(r) \geq 0$. Looking at a $D \in g^1_d$ we notice that $r(nD) \geq n$ and hence $c(r) \leq rc(1)$. From this I arrive at the guess that $c$ is always concave. In fact I guess that there exists some generic Clifford function $C_{max}$ and that we can characterize all Clifford functions as concave functions $0 \leq c \leq C_{max}$.

In my searches I have only been able to find results on special values of the Clifford function (for instance $c(1)$ corresponds to the gonality of a curve), but none on this function as an object. Is it possible to characterize all functions occurring as the Clifford function of some curve $C$?