# An irreducible germ of holomorphic function at origin is still irreducible around the origin?

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?

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Can any op link this question with this MSE post? –  Frank Science Apr 8 at 11:21