Map on finite Hausdorff space

Question

Let $X$and $Y$ be topological spaces where $Y$ is Hausdorff. Let $X ×Y$ be given the product topology. Then for a function $f : X\to Y$ which of the following statements are necessarily true?

1. if $f$ is continuous, then $\text{graph}(f)= \{(x, f(x)) | x \in X\}$ is closed in $X × Y $
2. if graph$(f)$ is closed in $X × Y $, then $f$ is continuous.
3. if graph $(f)$ is closed in $X × Y $ , then $f$ need not be continuous.
4. if $Y$ is finite, then $f$ is continuos.

By Closed graph theorem $1$ and $2$ should be true. How to approach for $4$ ?

Answer 1

Answer for 4). Let $E \subset X$, $E$ (non-empty and $E \neq X$) and $f(x)=1$ for $x \in E$, $0$ for $x \notin E$. Take $Y=\{0,1\}$. The $f$ is continuous iff $E$ is open and closed. So if $X$ is connected then $f$ cannot be continuous. So 4) is false.

Also, 2) is false (so 3) is true): let $f:\mathbb R \to \mathbb R $ be defined by $f(x)=\frac 1 x$ if $x \neq 0$ and $0$ if $x=0$. Then the graph of $f$ is closed but $f$ is not continuous.