  Consider a functional associating to a pair of compactly supported smooth
functions on a symplectic manifold the maximum of their Poisson bracket. We
show that this functional is lower semi-continuous with respect to the product
uniform (C^0) norm on the space of pairs of such functions. This extends
previous results of Cardin-Viterbo and Zapolsky. The proof involves theory of
geodesics of the Hofer metric on the group of Hamiltonian diffeomorphisms. We
also discuss a failure of a similar semi-continuity phenomenon for multiple
Poisson brackets of three or more functions.
