  We provide a $C^0$ counterexample to the Lagrangian Arnold conjecture in the
cotangent bundle of a closed manifold. Additionally, we prove a quantitative
$h$-principle for subcritical isotropic embeddings in contact manifolds, and
provide an explicit construction of a contact homeomorphism which takes a
subcritical isotropic curve to a transverse one. On the rigid side, we give
another proof of the Dimitroglou Rizell and Sullivan theorem \cite{RS22} which
states that Legendrian knots are preserved by contact homeomorphisms, provided
their image is smooth. Moreover, our method gives related examples of rigidity
in higher dimensions as well.
