show that p is not complete

**wopashui** · November 25th 2011, 05:45 PM

Let $D \subset R$ , and let f: D-->R be continuous. Prove that D is connected if { $x,f(x): x \in D$ }, the graph of f, is a connected subset of $R^2$

**Drexel28** · November 25th 2011, 07:57 PM

Originally Posted by wopashui

Let $D \subset R$ , and let f: D-->R be continuous. Prove that D is connected if { $x,f(x): x \in D$ }, the graph of f, is a connected subset of $R^2$

This is an if and only if. Let $\Gamma_f$ denote the graph, note then that $D=\pi_1(\Gamma_f)$ where $\pi_1:R^2\to R$ is the canonical projection onto the first coordinate--why does this tell us $D$ 's connected. Conversely, if $D$ is connected then $\Gamma_f=h(D)$ where $h: D\to R^2: x \mapsto (x,f(x))$ --why is $h$ continuous and why does this tell us that $\Gamma_f$ is connected?

**wopashui** · November 26th 2011, 08:35 AM

Originally Posted by Drexel28

This is an if and only if. Let $\Gamma_f$ denote the graph, note then that $D=\pi_1(\Gamma_f)$ where $\pi_1:R^2\to R$ is the canonical projection onto the first coordinate--why does this tell us $D$ 's connected. Conversely, if $D$ is connected then $\Gamma_f=h(D)$ where $h: D\to R^2: x \mapsto (x,f(x))$ --why is $h$ continuous and why does this tell us that $\Gamma_f$ is connected?

sorry, what is canonical projection ?

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