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The Brasselet number of a function $f$ with nonisolated singularitiesdescribes numerically the topological information of its generalized Milnorfibre. In this work, we consider two function-germs$f,g:(X,0)\rightarrow(\mathbb{C},0)$ such that $f$ has isolated singularity atthe origin and $g$ has a stratified one-dimensional critical set. We use theBrasselet number to study the local topology a deformation $\tilde{g}$ of $g$defined by $\tilde{g}=g+f^N,$ where $N\gg1$ and $N\in\mathbb{N}$. As anapplication of this study, we present a new proof of the L\^e-Iomdin formulafor the Brasselet number.

Local topology of a deformation of a function-germ with a one-dimensional critical set