I am looking at an old qualifying exam to study for my finals; it asks the following true/false question:
Let $f$ be a continuous, non-decreasing function defined on $[0,1]$, and let $E$ be a set of Lebesgue measure zero. Then, $f(E)$ is a set of Lebesgue measure zero.
I suspect this is false, but am not sure. Can anyone think of a solution without using the notion of absolute continuity? The reason for this is because AC won't be covered on the final.