  We show that a computable function $f:\mathbb R\rightarrow\mathbb R$ has
Luzin's property (N) if and only if it reflects $\Pi^1_1$-randomnes, if and
only if it reflects $\Delta^1_1(\mathcal O)$-randomness, and if and only if it
reflects $\mathcal O$-Kurtz randomness, but reflecting Martin-L\"of randomness
or weak-2-randomness does not suffice. Here a function $f$ is said to reflect a
randomness notion $R$ if whenever $f(x)$ is $R$-random, then $x$ is $R$-random
as well. If additionally $f$ is known to have bounded variation, then we show
$f$ has Luzin's (N) if and only if it reflects weak-2-randomness, and if and
only if it reflects $\emptyset'$-Kurtz randomness. This links classical real
analysis with algorithmic randomness.
