  We introduce the notion of a point on a locally closed subset of a symplectic
manifold being "locally rigid" with respect to that subset, prove that this
notion is invariant under symplectic homeomorphisms, and show that coisotropic
submanifolds are distinguished among all smooth submanifolds by the property
that all of their points are locally rigid. This yields a simplified proof of
the Humili\`ere-Leclercq-Seyfaddini theorem on the $C^0$-rigidity of
coisotropic submanifolds. Connections are also made to the "rigid locus" that
has previously been used in the study of Chekanov-Hofer pseudometrics on orbits
of closed subsets under the Hamiltonian diffeomorphism group.
