In Case of Oscillation Sequence of Newtonian Method

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?

• @Moo Added at OP. thx – Beverlie Jun 28 '17 at 13:20
• Which book? Perhaps it means cycles, which can happen in Newton's method. See math.stackexchange.com/a/89350/589. – lhf Jun 28 '17 at 13:22
• @Ihf this book written in Korean. How the ordinary books introduces the Cycles case? – Beverlie Jun 28 '17 at 13:23