Semicontinuity and X-rank

Let X ⊂ ℙr be an integral and non-degenerate variety. For each q ∈ ℙr, the X-rank rX(q) of q is the minimal cardinality of a subset of X whose linear span contains q. Obviously, rX-1(1) = X. In this note, we give conditions which assure that rX-1(2) ∪ X is not closed in ℙr and hence the function rX is not semicontinuous.