This question comes from Huybrechts's book *Complex Geometry, An Introduction*.
In proposition 1.1.35, the author claims that if $f$ is an irreducible holomorphic germ in $\mathcal{O}_{\mathbf{C}^n,0}$ at the origin of $\mathbf{C}^n$, then for any $z$ sufficiently close to the origin the holomorphic germ induced by $f$ in the local ring of $\mathbf{C}^n$ at z is irreducible.

But the proof only shows the claim holds on the complement of a thin subset.

**Question.** Is the claim true or false? Can anyone give an answer or a reference?