1. ## connected set

Hi.....

Let f:A------ R
is a continuous function and A is a connected set
prove that:
{ (x,f(x)):x in A}
is a connected set

2. You want to prove that $Z = f(A)$ is connected. Consider a continuous surjective map $l: A \to Z$. Let $Z = X \cup Y$ be a separation of $Z$ into 2 disjoint nonempty sets open in $Z$. Then $l^{-1}(X)$ and $l^{-1}(Y)$ are disjoint sets whose union is $A$. They are open and form a separation of $A$. Contradiction.

3. Thank you Particlejohn,

but I want to prove that :

$\{ (x,f(x)):x \in A\}
$
is a connected set in $R^2
$

4. Originally Posted by karimath
but I want to prove that :

$\{ (x,f(x)):x \in A\}
$
is a connected set in $R^2
$
The proof for this is very similar to what particlejohn suggested. Let $G = \{ (x,f(x)):x \in A\}$ and let G=X∪Y be a partition of G into open sets X and Y. Let $U = \{x\in A: (x,f(x))\in X\}$ and let $V = \{x\in A: (x,f(x))\in Y\}$. Then U, V are open and disjoint, and their union is A ... .

5. Thank you very much ,Opalg