A locally compact semitopological semigroup which is also a group is in fact a topological group, i.e. the existence of inverses plus separate continuity actually forces joint continuity.

This is a theorem of Ellis, see

R. Ellis, Locally compact transformation groups. Duke Math. J. 24 (1957) no. 2, 119–125

Quoting from the MathReview:

Let G be a group of homeomorphisms on a locally compact space X. Suppose G has a Hausdorff topology such that multiplication is continuous in each variable separately. If the function π:G × X → X defined by π(g,x)=g(x) is continuous on the left, the author shows that π is jointly continuous.

A locally compact semitopological semigroup which is also a group is in fact a topological group, i.e. the existence of inverses plus separate continuity actually forces joint continuity.

This is a theorem of Ellis, see

R. Ellis, Locally compact transformation groups. Duke Math. J. 24 (1957) no. 2, 119–125

Quoting from the MathReview:

Let G be a group of homeomorphisms on a locally compact space X. Suppose G has a Hausdorff topology such that multiplication is continuous in each variable separately. If the function π:G × X → X defined by π(g,x)=g(x) is continuous on the left, the author shows that π is jointly continuous.

EDIT: it's been noted in the comments that the part I quoted does not explain in any way why we can deduce continuity of inversion. In an earlier paper Ellis had shown that in a topological semigroup whose underlying semigroup is a group, inversion is continuous:

R. Ellis, A note on the continuity of the inverse. Proc. Amer. Math. Soc. 8 (1957), 372–373

A locally compact semitopological semigroup which is also a group is in fact a topological group, i.e. the existence of inverses plus separate continuity actually forces joint continuity. This is a theorem of Ellis, IIRC; I am out of the office right now but can look up some references tomorrow if required.

A locally compact semitopological semigroup which is also a group is in fact a topological group, i.e. the existence of inverses plus separate continuity actually forces joint continuity. 

This is a theorem of Ellis, see

R. Ellis, Locally compact transformation groups. Duke Math. J. 24 (1957) no. 2, 119–125

Quoting from the MathReview:

Let G be a group of homeomorphisms on a locally compact space X. Suppose G has a Hausdorff topology such that multiplication is continuous in each variable separately. If the function π:G × X → X defined by π(g,x)=g(x) is continuous on the left, the author shows that π is jointly continuous.