I am learning the basic concept of Newtonian method and text book introduces a function which newtonian step values are Oscillating as a example which is Newtonian method is inapplicable. However, I think for any kind of differentiable function at all domain, oscialltion guarantees the where the root is since if $x_n$ oscialltes between 1+h and 1-h where h>0, root would be 1.
Isn't it true?
example added.
$f(X) = \sqrt{x-1}, x\ge1$ and $-\sqrt{x-1}, x\le1$
In this case we can verify that the newton method keep oscillating about $x =1$
However, I think, Osciallation is good thing since we can just guarantee that that oscillating criterion becomes root.
Isnt't it?
