Singular points of weakly holomorphic functions Maciej P. Denkowski 1 IMUJ PREPRINT 2006/10 December 13th 2006 Abstract. In this paper we are interested in two kinds of singu- lar points of weakly holomorphic functions. Points where a weakly holomorphic function is not holomorphic and points at which it just is not continuous. The latter are closely connected to points of irreducibility of the given analytic set. We investigate the struc- ture of such points proving they form analytically constructible sets. We prove also that non-holomorphicity points of a given weakly or c-holomorphic function form an analytic subset of the singularities. 1. Introduction Throughout this paper A ⊂ C m is a locally analytic set. When trying to figure out which complex functions defined on A are the best generalization of the notion of a holomorphic function one comes across two natural notions. The first one is due to R. Remmert. Definition 1.1. (cf. [Wh]) A mapping f : A → C n is called c-holomorphic if it is continuous and the restriction of f to the subset RegA of regular points is holomorphic. We denote by O c (A, C n ) the ring of c-holomorphic mappings, and by O c (A) the ring of c-holomorphic functions. A well-known theorem states that a mapping defined in an open set is holo- morphic if and only if it is continuous and its graph is an analytic set (it is then a submanifold). We have a similar result for c-holomorphic mappings (cf. [Wh] 4.5Q), which motivates this generalization: 1 Krak´ ow 1991 Mathematics subject classification. 32B30 Key words and phrases. Complex analytic sets, weakly holomorphic functions, c- holomorphic functions, fibred products. Date: September 23rd 2006 1