Sorgenfrey line and continuous separating families

We study continuous separating families on linearly ordered extensions of the Sorgenfrey line S. Let R be the set of all real numbers, Z the set of all integers, and S∗=R×{n∈Z: n⩽0} with the lexicographical ordering ≼ and with the usual interval topology defined by ≼. Then S∗ is a linearly ordered extension of S. We prove that, in ZFC, S∗ does not admit a continuous separating families and that any linearly ordered extension of S does not admit a continuous separating family. Two problems posed by H. Bennett and D. Lutzer are answered.