# Square root: a contraction?

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!

• You have essentially answered your own question. What happens between $0$ and $1$? – Jonas Meyer Oct 23 '12 at 8:00
• But, for $x\ge 1$, it is a contraction. – Berci Oct 23 '12 at 8:52
• @Berci: For $x\geq \frac{1}{4}$ it is a contraction, too. – Jonas Meyer Oct 23 '12 at 15:33

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 $$\left|\frac14-0\right|\,?$$