Suppose V is a singular complex analytic curve inside \(\mathbb {C}^{2}\) . We investigate when a singular or non-singular complex analytic curve W inside \(\mathbb {C}^{2}\) with sufficiently small Hausdorff distance \(d_{H}(V, W)\) from V must intersect V. We obtain a sufficient condition on W which when satisfied gives an affirmative answer to our question. More precisely, we show the intersection is non-empty for any such W that admits at most one non-normal crossing type discriminant point associated with some proper projection. As an application, we prove a special case of the higher dimensional analog and also a holomorphic multifunction analog of a result by Lyubich and Peters (Geom. Funct. Anal. 24, 887–915 (2014)).
