I am trying to characterize the sensitivity of a function $f: R^N\to{}R$ to the perturbations in the input vector $\mathbb{x}=\left[x_1,\dots{}x_N\right]$. For that purpose, I evaluate Cramer-Rao bound for Gaussian i.i.d. arguments.

The function has a singularity at the point $\mathbb{x}_0$ where $f(\mathbb{x}_0)=0$, in the sense that $\|\nabla{}f\|\sim{}1/f$ as $\mathbb{x}\to\mathbb{x}_0$

The Cramer-Rao bound then doesn't make sense, because it diverges at $\mathbb{x}_0$, while the variance of $f$, obviously, remains bounded. What I am looking for, I guess, some type of "concentration of measure"/"deviation inequality"-type sharp bound on the variance of $f$.

The literature on concentration of measure phenomenon is extensive and deals with fairly advanced topics, whereas I am looking for something rather more basic. If you could point towards some starting point, your help will be appreciated.