When is the Kullback–Leibler divergence continuous?

Question

For two probability measures $P,Q$, if $P\ll Q$, then the Kullback–Leibler divergence (or the relative entropy) is defined as: $$D_{KL}(P||Q)=\int\log\bigg(\frac{dP}{dQ}\bigg)dP.$$ It is well known that for any two measures $P$ and $Q$, $D_{KL}$ is lower semicontinuous in $P,Q$. My question is: what are the conditions on $P,Q$ that make $D_{KL}$ continuous in the weak topology?