proof question 4

**wvlilgurl** · March 29th 2009, 05:45 AM

Suppose that f : R->R is conitnuous and that its image f(R) is bounded. Prove that there is a solution for the equation f(x)=x for x in R

**red_dog** · March 29th 2009, 07:10 AM

Let $m=\inf f(\mathbf{R}), \ M=\sup f(\mathbf{R})\Rightarrow m\leq f(x)\leq M, \ \forall x\in\mathbf{R}$

Let $g:\mathbf{R}\to\mathbf{R}, \ g(x)=f(x)-x$

$g(m)=f(m)-m\geq 0, \ g(M)=f(M)-M\leq 0$

g continuous $\Rightarrow \exists c\in[m,M]$ such as $g(c)=0\Rightarrow f(c)=c$

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