questions about connected and Hausdorff topological Space

Question

1: let $X$ be a Hausdorff topological Space , $Y \subseteq X $ is Nonempty and dense in $X$. let $f: X \longrightarrow X $ be a continuous functions so that $ \forall y \in Y, f(y) = y $.

Is $f$ a identical function?

My second question is:

2: Let $A$ be a connected subset of the topological space $X$and $Y \subseteq X $.

Can we say if $A \bigcap Y \neq \emptyset$ and $A \bigcap Y^{c}\neq \emptyset$, then $ A \bigcap \partial Y \neq \emptyset $? Why?

Answer 1

I don't know what you mean by "Nonsense", but $\{x\in X: f(x)=x\}$ is closed in $X$, so if it is dense it must be the whole space.

Answer 2

Yes to the second one, because for any $Y \subseteq X$, the three sets $\operatorname{int}(Y), \partial Y, \operatorname{int}(Y^c)$ are pairwise disjoint and partition $X$. If $A$ were to miss $\partial Y$ the other two open sets would disconnect $A$, which cannot be.