  We study the nature of applicative bisimilarity in $\lambda$-calculi endowed
with operators for sampling from continuous distributions. On the one hand, we
show that bisimilarity, logical equivalence, and testing equivalence all
coincide with contextual equivalence when real numbers can be manipulated only
through continuous functions. The key ingredient towards this result is a novel
notion of Feller-continuity for labelled Markov processes, which we believe of
independent interest, being a broad class of LMPs for which coinductive and
logically inspired equivalences coincide. On the other hand, we show that if no
constraint is put on the way real numbers are manipulated, characterizing
contextual equivalence turns out to be hard, and most of the aforementioned
notions of equivalence are even unsound.
