On the Analyticity of Certain Singularity Sets

In this note we prove the

(1.1) Theorem. Let W be an open set in Cⁿ⁺¹ and A⊂W a closed subset such that

(a) there exists a function f holomorphic on W - A which is singular at every point of A,

(b) the projection mapping Cⁿ⁺¹ → Cⁿ is discrete on A.

Then A is an analytic set in W.