A celebrated theorem due to Gromov and Eliashberg states that the C^0-limit of a se­quence of sym­plec­to­mor­phisms is sym­plec­tic (if smooth). This rigid­ity phe­nom­e­non mo­ti­vated the study of C^0 sym­plec­tic geom­e­try which is con­cerned with con­tin­u­ous analogs of clas­si­cal notions. In joint works with V. Hu­milière and S. Sey­fad­dini, we showed that coisotropic submanifolds together with their characteristic foliations are also C^0 rigid. I will explain this result (and some consequences) and in particular how it relies on a continuous analog of a dynamical property satisfied by coisotropics which generalizes a foundational theorem in C^0-Hamiltonian dynamics.