  We prove a new variant of the energy-capacity inequality for closed rational
symplectic manifolds (as well as certain open manifolds such as cotangent
bundle of closed manifolds...) and we derive some consequences to
C^0-symplectic topology. Namely, we prove that a continuous function which is a
uniform limit of smooth Hamiltonians whose flows converge to the identity for
the spectral (or Hofer's) distance must vanish. This gives a new proof of
uniqueness of continuous generating Hamiltonian for hameomorphisms. This also
allows us to improve a result by Cardin and Viterbo on the C^0-rigidity of the
Poisson bracket.
