# connected set

• Aug 12th 2008, 10:11 PM
karimath
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
• Aug 12th 2008, 10:55 PM
particlejohn
You want to prove that $\displaystyle Z = f(A)$ is connected. Consider a continuous surjective map $\displaystyle l: A \to Z$. Let $\displaystyle Z = X \cup Y$ be a separation of $\displaystyle Z$ into 2 disjoint nonempty sets open in $\displaystyle Z$. Then $\displaystyle l^{-1}(X)$ and $\displaystyle l^{-1}(Y)$ are disjoint sets whose union is $\displaystyle A$. They are open and form a separation of $\displaystyle A$. Contradiction.
• Aug 13th 2008, 02:12 AM
karimath
Thank you Particlejohn,

but I want to prove that :

$\displaystyle \{ (x,f(x)):x \in A\}$ is a connected set in $\displaystyle R^2$
• Aug 13th 2008, 07:51 AM
Opalg
Quote:

Originally Posted by karimath
but I want to prove that :

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

The proof for this is very similar to what particlejohn suggested. Let $\displaystyle G = \{ (x,f(x)):x \in A\}$ and let G=X∪Y be a partition of G into open sets X and Y. Let $\displaystyle U = \{x\in A: (x,f(x))\in X\}$ and let $\displaystyle V = \{x\in A: (x,f(x))\in Y\}$. Then U, V are open and disjoint, and their union is A ... .
• Aug 13th 2008, 11:23 AM
karimath
(Clapping) Thank you very much ,Opalg