Show That a Function is Contractive?

[DeadxBunny]

Homework Statement

Show that the following function is contractive on the indicated intervals. Determine the best values of [lamda] in Equation (2).

abs(x)^(2/3) on abs(x) < or = 1/3

Homework Equations

A mapping (or function) F is said to be contractive if there exists a number [lamda] less than 1 such that:

(Equation (2))
abs(F(x)-F(y)) < or = [lamda]*abs(x-y)

for all points x and y in the domain F.

The Attempt at a Solution

I'm not really sure what to do with this one or how to get [lamda] in this case.

 

[tim_lou]

mean value theorem?

it seems that for the function that is given, such lamda doesn't exist. consider a interval extremely close to 0, let's say
$$(-\epsilon, \epsilon)$$

you see that
$$\frac{|F(x)-F(y)|}{|x-y|}$$
goes to infinity, as it approaches the derivative at 0.

are you sure you have the correct question?

 

[DeadxBunny]

Thanks for responding, tim_lou. Yes, I'm sure I have the correct question. :) Maybe it's a trick question and it's actually not contractive?