  We investigate the role of continuous reductions and continuous
relativisation in the context of higher randomness. We define a higher analogue
of Turing reducibility and show that it interacts well with higher randomness,
for example with respect to van-Lambalgen's theorem and the Miller-Yu / Levin
theorem. We study lowness for continuous relativization of randomness, and show
the equivalence of the higher analogues of the different characterisations of
lowness for Martin-L\"of randomness. We also characterise computing higher
$K$-trivial sets by higher random sequences. We give a separation between
higher notions of randomness, in particular between higher weak-2-randomness
and $\Pi^1_1$-randomness. To do so we investigate classes of functions
computable from Kleene's~$O$ based on strong forms of the higher limit lemma.
