  The Brasselet number of a function $f$ with nonisolated singularities
describes numerically the topological information of its generalized Milnor
fibre. In this work, we consider two function-germs
$f,g:(X,0)\rightarrow(\mathbb{C},0)$ such that $f$ has isolated singularity at
the origin and $g$ has a stratified one-dimensional critical set. We use the
Brasselet 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 an
application of this study, we present a new proof of the L\^e-Iomdin formula
for the Brasselet number.
