Function

**lehder** · September 26th 2009, 09:50 AM

Hi everybody,

f is a function defined from [a;b] to [a;b], such as:

$(\forall (x;t)\in[a;b]^2) |f(x)-f(t)|<|x-t|$

1)- i must show that f is continuous on [a;b]

2)-And i must show also that f accepts a fixed point on [a;b]

Can you help me please?

And thank you anyway.

**red_dog** · September 26th 2009, 12:39 PM

1) Let $x_0\in[a,b]$ . Then $|f(x)-f(x_0)|<|x-x_0|$

$0\leq\lim_{x\to x_0}|f(x)-f(x_0)|\leq\lim_{x\to x_0}|x-x_0|=0\Rightarrow$

$\Rightarrow\lim_{x\to x_0}(f(x)-f(x_0))=0\Rightarrow\lim_{x\to x_0}f(x)=f(x_0)$

Then the function is continuous.

2) Let $g(x)=f(x)-x, \ g:[a,b]\to\mathbb{R}$

$g(a)=f(a)-a\geq 0, \ g(b)=f(b)-b\leq 0$

But g is continuous, so exists $c\in[a,b]$ such as $g(c)=0\Rightarrow f(c)=c$ .

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