Towards the C^0 flux conjecture

In this note, we generalise a result of Lalonde, McDuff and Polterovich concerning the $C^0$ flux conjecture, thus confirming the conjecture in new cases of a symplectic manifold. Also, we prove the continuity of the flux homomorphism on the space of smooth symplectic isotopies endowed with the $C^0$ topology, which implies the $C^0$ rigidity of Hamiltonian paths, conjectured by Seyfaddini.