  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-Peters.
