  Let $(X,0)$ be an isolated hypersurface singularity defined by
$\phi\colon(\mathbb C^n,0)\to(\mathbb C,0)$ and $f\colon(\mathbb
C^n,0)\to\mathbb C$ such that the Bruce-Roberts number $\mu_{BR}(f,X)$ is
finite. We first prove that
$\mu_{BR}(f,X)=\mu(f)+\mu(\phi,f)+\mu(X,0)-\tau(X,0)$, where $\mu$ and $\tau$
are the Milnor and Tjurina numbers respectively of a function or an isolated
complete intersection singularity. Second, we show that the logarithmic
characteristic variety $LC(X,0)$ is Cohen-Macaulay. Both theorems generalize
the results of a previous paper by some of the authors, in which the
hypersurface $(X,0)$ was assumed to be weighted homogeneous.
