Continuity in topological groups

We prove that, when G is a group equipped with a Baire and metrizable topology, if there is a second category dense subset S of G such that the right translations rho(s) and rho(s-1) are continuous for all s is an element of S and each left translation lambda(s) is an element of G, is almost-continuous (defined below) on a residual subset of G, then G is a topological group. Among other consequences, this yields that when G is a second countable locally compact right topological group, its topological centre is a topological group. (c) 2005 Elsevier B.V. All rights reserved.