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The graph of $f$ is $G(f) = \{(x,f(x)) : x\in X\} \subseteq X\times Y$

$X$ and $Y$ are metric spaces. $Y$ is compact.

$f$ is continuous iff $G(f)$ is a closed set.

I got the closest answer here but I tried it by myself first and got stuck at one point and I need help on that particular situation which I didn't get anywhere else/

$\Rightarrow$ part: Let $(z_n)=(x_n,f(x_n))\in G_f$ be a convergent sequence of $G(f)$. If $(x,y)$ is its limit. We have to show that $y=f(x)$ in other words $(x,y)\in G_f$.

$x_n \to x$ $\Rightarrow$ $f(x_n)\to f(x)$[By continuity of $f$.] $\Rightarrow f(x)=y$ by uniqueness of the limit. Hence $G_f$ is closed.

$\Leftarrow$ part: Let $x\in X$ and $(x_n)$ a convergent sequence with limit $x$. You have to prove that $(f(x_n))$ is convergent in $Y$ with limit $f(x)$. I have used the sequence $z_n=(x_n,f(x_n))$ and $G_f$ is closed in the compact space $Y$ and hence $G_f$ is compact. Then there is subsequence $(x_{n_k},f(x_{n_k})) \to (x,y)\in G_f$. Then we will have $y=f(x)$ but how do I prove that $f(x_n) \to f(x)$? It is true that every subsequence of $f(x_n)$ has a subsequence converging to $f(x)$.

Ri-Li
  • 9,038

1 Answers1

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From the comment I got my answer which comes from this lemma:

Lemma Let $Y$ be a compact metric space and $(y_n)$ a sequence whose terms belong to $Y$. If every convergent subsequence of $(y_n)$ converges to the same limit $\ell\in Y$, then $(y_n)$ converges to $\ell$.

Proof Suppose the contrary. Then, there exists $\epsilon>0$, such that :

$$\forall N\in\mathbb{N},\,\exists n\ge N;\,d(y_n,\ell)>\epsilon$$

This allows us to construct a subsequence $(y_{n_k})$ such that :

$$\forall k\in\mathbb{N},d(y_{n_k},\ell)>\epsilon$$

Now extract from $(y_{n_k})$ a convergent subsequence : its limit $\ell$ from the hypothesis and hence we get $0=d(\ell,\ell)\ge\epsilon$ ...

A contradiction!

Now someone can close this answer but I can keep it under my record and if someone will proceed in this way. They will get help from it. I asked the question because I was checking one of the obvious ways that can come to our mind. Thanks a lot!

Ri-Li
  • 9,038