We present in this paper sufficient conditions for the topological triviality of families of germs of functions defined on an analytic variety $V$. The main result is an infinitesimal criterion based on a convenient weighted inequality, similar to that introduced by T. Fukui and L. Paunescu. When $V$ is a weighted homogeneous variety, we obtain as a corollary, the topological triviality of deformations by terms of non negative weights of a weighted homogeneous germ consistent with $V$. Application of the results to deformations of Newton non-degenerate germs with respect to a given variety is also given.
"Topological triviality of families of functions on analytic varieties." Nagoya Math. J. 175 39 - 50, 2004.