Assume that $G$ is a compact group acting by isometries on a (compact) Riemannian manifold (M,g), with principal orbits of dimension $d>0$. For $x\in M$, let $G(x)$ denote the $G$-orbit of $x$, by $G_x$ the stabilizer of $x$ and by $G_x^0$ the identity connected component of $G_x$. Does anyone know a reference where it it proved the the function $f(x)=\vert G_x/G_x^0\vert\cdot vol_d(G(x))$ is continuous on $M$? Here, $\vert .\vert$ denotes the cardinality, and $vol_d$ the $d$-dimensional volume (induced by the restriction of the metric $g$ on the orbit).

How about a pseudo-Riemannian extension of the above? If $(M,g)$ is pseudo-Riemannian, then one defines $f(x)$ as above when $G(x)$ is a nondegenerate submanifold, and $f(x)=0$ otherwise. Is such $f$ continuous?

Note that $f(x)\ne 0$ only if $G(x)$ has dimension $d$, i.e., if $G(x)$ is either a principal or an exceptional orbit. The function $v(x)=vol_d(x)$ is only continuous at points $x$ whose orbit is principal, but it fails to be continuous at points with exceptional orbit.