Let $G$ be a connected reductive group acting on a projective variety $X$. Let $L$ be a $G$-linearized (very) ample line bundle on $X$, and let $R$ be the ring of sections (or homogeneous coordinate ring of $(X, L)$). Let $R_k = H^0(X, L^{\otimes k})$ denote the $k$-th graded piece of $R$, and $R_{k, \lambda}$ its $\lambda$ isotypic component where $\lambda$ is a dominant weight.

Consider the GIT quotient $X //_\lambda G$ i.e. $Proj(\bigoplus_k R_{k, k\lambda})$. Then one can define the volume of this GIT quotient $f(\lambda) = vol(X //_\lambda G)$ say as the leading coefficient of the Hilbert polynomial of the graded algebra $\bigoplus_k R_{k, k\lambda}$ times $d!$ where $d = dim(X //_\lambda G)$. By homogeneity one can extend f to rational values of $\lambda$. (When $X$ is smooth $f$ is just the Duistermaat-Heckman function in symplectic geometry on the moment/Kirwan polytope.)

Question: Under what conditions is f continuous at 0, i.e. as \lambda approaches $0$, $f(\lambda)$ approaches $f(0)$ (we consider $f$ as a function on the moment polytope and we assume that $0$ in on the boundary of the moment polytope)? Little bit weaker questions is: when can we find a sequence of regular dominant $\lambda_i$ approaching $0$ such that $f(\lambda_i)$ approaches $f(0)$?

Clearly a sufficient condition for continuity of $f$ is that $X$ is smooth and $0$ is a regular value of the moment map.

For example, is $f$ continuous under the assumption that generic orbits of the action of $G$ on $Spec(R)$ (cone over $X$) are closed?