> 
> Let $f:\mathbb R\to \mathbb R$ be function .  
> 
>  Define $2$ sets as A and B in $\mathbb R^2 $ as follows:  
> 
>  A={$(x,y)|y<f(x)$},B={$(x,y)|y>f(x)$} .  
> 
>  Then $f$ is continous on $\mathbb R$ iff A and B are open subset of $\mathbb R^2$
>    
> 
> 
> 
> 

$\to $ to show A open that means for any $(a\_1,a\_2)\in A$ There exist some $r>0$ ball which contain in A .i.e $K=B\_r((a\_1,a\_2))\subset A$  

$(m,n)\in K$
As f is continous on R$\forall \epsilon >0,\exists \delta >0 $ such that $\forall x\in R$ $|x-y|<\delta$ then $|f(x)-f(y)|<\epsilon$ 

$f(a\_1)>a\_2$  

I could not get idea to proceed further for any part.
Actually I don't able to understand how is this set look like.And How to use continuty assumption  

Any help will be appreciated