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- TL;DR Summary
- Want to understand which kind of functions I can be sure that the method of fixed-point iteration is suitable

Hi PF

Not every function works when we try to compute the root with this method

The following theorem guarantees that the method of fixed point iteration will work for a particular class of functions

Suppose that ##f## is defined on an interval ##I=[a,b]## and satisfies the following two conditions

(i) ##f(x)## belongs to ##I## whenever ##x## belongs to ##I##; and

(ii) there exists a constant ##K## with ##0<K<1## such that for every ##u## and ##v## in ##I##

$$|f(u)-f(v)|\leq K|u-v|$$

Then ##f## has a unique fixed point ##r## in ##I##, that is, ##f(r)=r##, and starting with any number ##x_0## in ##I##, the iterates

$$x_1=f(x_0),\qquad{x_2=f(x_1),...}$$

converge to ##r##

1- Condition (ii) of theorem implies that ##f## is continuous on ##I=[a,b]##. Use condition (i) to show that ##f## has a unique fixed point ##r## on ##I##. Apply the Intermediate-Value Theorem to ##g(x)=f(x)-x## on ##[a,b]##.

2- Use condition (ii) of theorem and mathematical induction to show that ##|x_n-r|\leq K^{n}|x_0-r|##. Since ##0<K<1##, we know that ##K^{n}\rightarrow{0}## as ##n\rightarrow{\infty}##. This shows that ##\lim_{n\rightarrow{infty}}{x_n=r}##

Thanks

Not every function works when we try to compute the root with this method

The following theorem guarantees that the method of fixed point iteration will work for a particular class of functions

**A fixed point theorem**Suppose that ##f## is defined on an interval ##I=[a,b]## and satisfies the following two conditions

(i) ##f(x)## belongs to ##I## whenever ##x## belongs to ##I##; and

(ii) there exists a constant ##K## with ##0<K<1## such that for every ##u## and ##v## in ##I##

$$|f(u)-f(v)|\leq K|u-v|$$

Then ##f## has a unique fixed point ##r## in ##I##, that is, ##f(r)=r##, and starting with any number ##x_0## in ##I##, the iterates

$$x_1=f(x_0),\qquad{x_2=f(x_1),...}$$

converge to ##r##

**Hints for the proof**1- Condition (ii) of theorem implies that ##f## is continuous on ##I=[a,b]##. Use condition (i) to show that ##f## has a unique fixed point ##r## on ##I##. Apply the Intermediate-Value Theorem to ##g(x)=f(x)-x## on ##[a,b]##.

2- Use condition (ii) of theorem and mathematical induction to show that ##|x_n-r|\leq K^{n}|x_0-r|##. Since ##0<K<1##, we know that ##K^{n}\rightarrow{0}## as ##n\rightarrow{\infty}##. This shows that ##\lim_{n\rightarrow{infty}}{x_n=r}##

**First**: ¿Why bounded Newton quotient means continuity on the interval?.Thanks