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.
Conway's base 13 function, see e.g. here for its definition, is discontinuous everywhere and has the property that it assumes every real value on every open interval, however small.
I think it fits all your requirements.