The question is as follows: Let $f:[0,1] \rightarrow R$ be a non-decreasing continuous function, and let $A \subset [0,1]$ be a set of Lebesgue measure $0$.
(a)Suppose that $f$ satisfies the following condition: (*) For some constant $C \geq 0$, $|f(s)-f(t)| \leq C|s-t|$ for all $s,t \in [0,1].$ Prove that $f(A)$ has measure zero.
This part is ok, I had no problem proving that. But part (b) has been giving me some issues:
(b) Suppose that $f$ does not satisfy condition (*). Does the conclusion in part (a) still hold?
Me and my office mate feel that if the lipschitz condition does not hold, then $f(A)$ will not have measure zero. We have tried to come up with counterexamples, but are having problems defining a continuous function on a set of measure 0.