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I understand that to use Fixed Point Iteration we demand that the function will be continues and will both the range and domain will be in the same closed interval and that $|f'(x)|<1$ for all $x$ in the interval.

Do we demand that the function have opposite signs too?

newhere
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  • Note that the function $g(x) = x - f(x)$ has $g'(r) \not = 0$ at the fixed point $r$ of $f$. It follows from the definition of differentiability that $g$ (rather than $f$) changes sign near $x=r$. – Carl Christian Jan 02 '20 at 10:49

1 Answers1

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The (constant) function $f(x)=1$ has a fixed point at $x=1$ without being negative anywhere.

You want opposite signs if you need to find a root of the function.

Also, if the function is continuous and its range falls inside its domain, it already has a fixed point. You want $|f'(x)|<1$ around the fixed point if the fixed-point iteration is to converge.

Lutz Lehmann
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