# Thread: closed sets and continuity

1. ## closed sets and continuity

Let $f:R \rightarrow R$ be any 1-variable function. Define $A=\{(x,y)\in R^2:y\leq f(x)\}$ and $B=\{(x,y) \in R^2 : y\geq f(x)\}$.
Prove that $f$ is continuous $iff$ $A$ and $B$ are closed in $R^2$.

2. Use the sequence definitions! Keep in mind a sequence of coordinate pairs converges if and only if the individual coordinates converge, and relate that to the convergence-preserving properties of continuous functions.

3. ## closed set

where do you want a sequence to converge?

4. Originally Posted by Kat-M
where do you want a sequence to converge?
Formally, here are the two qualities I was talking about:

Let $(x_n,y_n)$ be a sequence of points in $\mathbb{R}^2$. Then $(x_n,y_n)\rightarrow (x,y)$ if and only if $x_n\rightarrow x$ and $y_n\rightarrow y$.

Let $f:\mathbb{R}\rightarrow\mathbb{R}$. Then $f$ is continuous at $x$ if and only if for every sequence $x_n\rightarrow x$, $f(x_n)\rightarrow f(x)$.