Suppose that $X \subset \mathbb{CP}^n$ is a (possibly singular) algebraic curve. Let $H_{\tau} \subset \mathbb{CP}^n$ be a family of hyperplanes, with $\tau \in (-1,1)$.
By Bezout's theorem, the count of intersection points (with multiplicity) is independent of $\tau$. (Let us assume that $H_{\tau}$ does not contain any irreducible components of $X$, and that $X$ is reduced. This justifies the application of Bezout's theorem).
Question: I would like to say that the intersection points vary continuously in $\mathbb{CP}^n$, with respect to the Euclidean topology. Is this statement true, and if so, where can it be found in the literature?