I am looking for proof of $+$ and $\times$ are continuous operations without using the standard definition of continuity (1. $\epsilon-\delta$, or 2. preimage of open sets or 3. sequential continuity) Any other proof would be considered valid for this purpose
Define: Addition and multiplication as mappings
$+: \mathbb{R} \times \mathbb{R} \to \mathbb{R}, (a,b) \mapsto a+b$
$\times: \mathbb{R} \times \mathbb{R} \to \mathbb{R}, (a,b) \mapsto a \times b$
Using obscure definition of continuity:
$f$ is continuous if $\forall A \subset \mathbb{R} \times \mathbb{R}, f(\overline A) \subseteq \overline {f(A)}$
We know that singletons are closed in $\mathbb{R}$, closed sets are closed under cartesian product, so $A = \{a\} \times \{b\} \subset \mathbb{R} \times \mathbb{R}$ is closed
$f(\{a\} \times \{b\}) = f(\overline{\{a\} \times \{b\}}) = \overline{f(\{a\} \times \{b\})}$
The latter equality because $f(\{a\} \times \{b\})$ is a singleton, so $f$ is continuous
Can someone please check on the validity? And perhaps offer an alternative proof, thanks
Here's an easy way to tell that something must be wrong with your proof: what sort of assumptions on $f$ does it use? Your proof doesn't assume anything about $f$, so - if it were valid - would prove that every function is continuous.
The flaw, as the comments have pointed out, is that you need to show $$f(\overline{A})\subseteq\overline{f(A)}$$ for every $A\subseteq \mathbb{R}^2$, not just those of the form $\{a\}\times\{b\}$. Here's an example which might make this easier to visualize (on $\mathbb{R}$, instead of $\mathbb{R}^2$, for simplicity):
Let $f(x)$ be the characteristic function of $\mathbb{Q}$: $1$ if $x$ is rational, $0$ if $x$ is not rational. (This is called the Dirichlet function.)
Take $A=\mathbb{Q}$. What is $\overline{A}$? What is $f(\overline{A})$?
On the other hand, what is $f(A)$? What is $\overline{f(A)}$?
In terms of using this definition to prove things, I think it's generally faster to first prove that this is equivalent to the $\epsilon$-$\delta$ definition, and then use that one; in most cases that makes things a little more concrete and easy to figure out (at least early on). Of course, Your Mileage May Vary.