Even continuity and topological equicontinuity in topologized semigroups

A topologized semigroup X having an evenly continuous resp., topologically equicontinuous, family R X of right translations is investigated. It is shown that: (1) every left semitopological semigroup X with an evenly continuous family R X is a topological semigroup, (2) a semitopological group X is a paratopological group if and only if the family R X is evenly continuous and (3) a semitopological group X is a topological group if and only if the family R X is topologically equicontinuous. In particular, we get that for any paratopological group X which is not a topological group, the family R X provides an example of a transitive group of homeomorphisms of X that is evenly continuous and not topologically equicontinuous. The last conclusion answers negatively a question posed by H.L. Royden.