connected set

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August 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
August 12th 2008, 10:55 PM
particlejohn

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.
August 13th 2008, 02:12 AM
karimath

Thank you Particlejohn,

but I want to prove that :

$\{ (x,f(x)):x \in A\} <br />$ is a connected set in $R^2 <br />$
August 13th 2008, 07:51 AM
Opalg

Quote:

Originally Posted by karimath

but I want to prove that :

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

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 ... .
August 13th 2008, 11:23 AM
karimath

(Clapping) Thank you very much ,Opalg

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