Properties of Topological Groups I'm working though William Basener's Topology and Its Applications and I have come across a problem I can't solve.  The book defines a topological group as a group equipped with a topology where for each element $a$, $L_{a} (x) = a + x$ and $R{a}(x) = x + a$ are both continuous.  I need to prove that if the topology underlying the group is Hausdorff then $f(x, y) = x - y$  is continuous iff those all such functions $L$ and $R$ are continuous.  Any ideas?

The definition of topological group which you wrote is highly non-standard (I never saw it before) and the claim which you tried to prove is false. The situation is described at the first page of my paper. :-)
