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 ρ s and ρ s −1 are continuous for all s ∈ S and each left translation λ s , s ∈ 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.