Let $X$ be infinite set with cocountable topology
Let Y be topological space and $f:X \to Y$ and $x_n \to x$ show $f(x_n)=f(x)$
extra: does this mean it is countinous
Defintions
Given an uncountable set $X$, the co-countable topology on $X$ is
$$T_{\text{co-countable}} =\{ A \subset X : A^c \cap X \text{ is countable }\} \cup \{ \emptyset \}.$$
Let $(X,T)$ be a topological space. If $(x_n)_{n=1}^\infty$ is a sequence in $X$ we say it converges to $x_0 \in X$, $x_n \to x_0$,
if $\forall $ open set $U \subset X$ s.t $x_0 \in U$, $\exists$ $n_0 \in \mathbb N$ s.t $x_n \in U$ for all $n \geq n_0$.
Def of $f(x_n) =f(x)$
Let $(Y,T_2)$ be a topological space. If $(f(x_n))_{n=1}^\infty$ is a sequence in $Y$ we say it converges to $f(x) \in X$, $f(x_n) \to f(x_0)$,
if $\forall $ open set $V \subset Y$ s.t $f(x_0) \in V$, $\exists$ $n_0 \in \mathbb N$ s.t $f(x_n) \in V$ for all $n \geq n_0$.
missing of defigion of mapping in a topological spaece
I am trying to tied it toghter like its real. but dont see it
Let $(x_n)$ be a sequence in $X$ in the co-countable topology, that converges to $x$. I claim that the sequence is eventually constantly $x$: $\exists n_0: \forall n \ge n_0: x_n = x$
To this end define $N = \{x_n: n \in N\} \setminus \{x\}$. This set is at most countable and $x \notin N$, so $U = X\setminus N$ is open in the co-countable topology and $x \in U$. Then by the definition of convergence $\exists n_0: \forall n \ge n_0: x_n \in U$. This $n_0$ is as required as $x_n \in U$ implies $x_n = x$ (all $x_n \neq x$ are in $N$, so not in $U$).
So this implies that if $x_n \rightarrow x$ ,then $f(x_n) \rightarrow f(x)$, as the image of an eventually constant-$x$ sequence is also eventually constant-$f(x)$ hence convergent to $f(x)$ in any topology.
Conclusion: any function defined on the co-countable topology is sequentially continuous, but not necessarily continuous. Consider the identity to a space with a finer topology: so $f(x) =x $ from $X$, cocountable to $X$ discrete (for uncountable $X$) is not continuous but it is sequentially continuous.
This is analogous to the fact that any function on a discrete space is continuous.
Define $N=\lbrace x_n\rbrace_{n\in\mathbb{N}}$ and suppose $x\notin N$. Then taking $U=X\setminus N,x\in U\in T$ and $\forall n\in\mathbb{N},x_n\notin U$ so $x_n\not\rightarrow x$. Therefore $x\in N$, so $\exists n_0\in\mathbb{N},x=x_{n_0}$, so $f(x)=f(x_{n_0})$.
The question is unclear so I don’t know if until this point I answered it.
If the answer is what you searched, then $f$ isn’t necessary continuous.