I know there are a lot of strange functions $f~:~\mathbb R \to \mathbb R$.

I'm looking for an "elementary but complete" exposition of a result discovered by W. Sierpi\'nski and A. Zygmund in "Sur une fonction qui est discontinue sur tout ensemble de puissance du continu." *Fund. Math.*, vol. 4, pp.316–318, 1923 stating (in simple form) that there exist a function $f~:~\mathbb R \to \mathbb R$ such that for every non empty open interval $I$, $f(I)=\mathbb R$ .

And also, a probably well-known fact for experts in real analysis: If an arbitrary continuous function $f~:~ [0,1] \to \mathbb R$ is given, is it true that $f_{|D}$ is monotonic on some dense set $D\subset [0,1]$ or some set $D$ of positive measure?

It seems to be true if I replace "dense set" by "perfect subset" (according to Jack Brown). If this is not true an example is highly appreciated.