Say $E \subset [0,1]$ is a null set. Let $f: [0,1] \rightarrow [0,1] $. Do you think $f(E)$ is a null set or not? Just being curious.
(DEF): A set $A$ is null if given any $\epsilon > 0$, there exists a sequence of intervals $\{I_n\}_{n\geq1}$ such that
$$ A \subseteq \bigcup _{n=1}^{\infty}I_n$$ and $$ \sum |I_n| < \epsilon $$
if $f$ is continuous, is $f(E)$ nullset or not?