# Existence of entirely discontinuous and entirely surjective funtions

Is there a function $$f : [0,1] \rightarrow [0,1]$$ such that for all dense subsets S of $$[0,1]$$ $$f$$ is discontinuous for all points in that subset? Could you give an example?

Is there a function $$f : [0,1] \rightarrow [0,1]$$ such that for all $$(a,b)$$ $$f$$ defined in $$(a,b)$$ is surjective on $$[0,1]$$? What if the codomain of f is $$\mathbb{R}$$? Could you give an example?

I am also interested into learning if this is true for $$(0,1)$$ and if these properties have a name rather than what I call "entirely" discontinuous/surjective.

edit: from what I understand conway's 13 is "purely" surjective but is not "purely" discontinuous. Since for every $$(a,b)$$ there is a $$c \in (a,b)$$ s.t. $$f(c) = 0$$ hence on the set of all $$c$$ (which is dense), $$f$$ is continuous.

• Do you know Conway's 13? – Asaf Karagila May 7 at 13:04
• this question is phrased very incorrectly. You have to move quantifiers around – mathworker21 May 7 at 13:09
• I'd recommend looking at some of the questions listed under "Related". – Gerry Myerson May 7 at 13:28