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

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- Aug 12th 2008, 10:11 PMkarimathconnected 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 PMparticlejohn
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 AMkarimath
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 AMOpalg
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 AMkarimath
(Clapping) Thank you very much ,Opalg