How do you know if a function is a contraction or a shrinking map and how do you find a unique point?

Question

Let $f(x) = (1/2)|x|$

a). What is the value of $f′(x)$ on $(−∞,0)$?

b). What is the value of $f′(x)$ on $(0,∞)$?

c). What is the value of $|f′(x)|$ on $(−∞, 0) ∪ (0, ∞)$

d). By part c). we can conclude that $|f(x)−f(y)|/|x-y| ≤ 1/2$ for all $x,y ∈ R$. So can you conclude $f$ is a contraction?

e). Is there a unique fixed point of $f$? If so, what is it?

Answer 1

Partial answers and hint:

You have $f(x)-f(y)=\frac{1}{2}\left( |x|-|y| \right)$ then $$|f(x)-f(y)|=\frac{1}{2}||x|-|y||\le \frac{1}{2}|x-y|,$$ so $f$ is a contractive map.

Also observe that $f(0)=0$.

For $x>0$ you have $f(x)=\frac{x}{2}$ and $f'(x)=\frac{1}{2}$.