$f(x) = x$

Suppose that $f: [0,1] \rightarrow [0,1]$ is a continuous function. Show that is has a fixed point, i.e there is a real number $x$ such that $f(x) = x$ .

This is a list of Calculus proof based problems that I like. Please avoid posting complete solutions, so that others can work on it.

#Calculus #Proofs

Easy Math Editor

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`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
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`\sum_{i=1}^3`	$\sum_{i=1}^3$
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Comments

On $[0,1]$ , consider : $g(x)=f(x)-x$ , then : $g(0)=f(0)\geq 0$ , and $g(1)=f(1)-1\leq 0$ , because : $0\leq f(x)\leq 1$ .

And note that $g$ is continuous since it is the sum of two contisinuous functions. Therefore it has a root in $[0,1]$ by the IVT. Which is equivalent with saying that there is some $x\in [0,1]$ such that $x=f(x)$ .

Haroun Meghaichi - 7 years, 2 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

f(x)=x f(x) = x f(x)=x

Comments

$f(x) = x$