Is the square root a contraction? If yes, there should be only one fixed point; yet, there are two of them, 0 and 1! Thanks for help!
Square root: a contraction?
3
$\begingroup$
real-analysis
-
1You have essentially answered your own question. What happens between $0$ and $1$? – 2012-10-23
-
1But, for $x\ge 1$, it *is* a contraction. – 2012-10-23
-
1@Berci: For $x\geq \frac{1}{4}$ it is a contraction, too. – 2012-10-23
1 Answers
6
You’ve actually answered your own question: it’s not a contraction. I suspect, however, that what you’d really like is an example showing that it’s not a contraction. If $f(x)=\sqrt x$, how does $$\left|f\left(\frac14\right)-f(0)\right|$$ compare with $$\left|\frac14-0\right|\,?$$