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 $D\subset [0,1]$?

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.

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.

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 $D\subset [0,1]$?

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.

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 $D\subset [0,1]$?

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.