10.8 Fixed points[2C8]

E296

[0TG]Find a function \(f:ℝ→ℝ\) such that

\[ ∀ x,y∈ ℝ, x≠ y⇒ |f(x)-f(y)| {\lt} |x-y| \]

but that has no fixed points. Hidden solution: [UNACCESSIBLE UUID ’27D’]

E296

[0TH]Prerequisites:9.Let \((X,d)\) be a compact metric space, and let \(f:X→ X\) be such that

\[ ∀ x,y∈ X, x≠ y⇒ d(f(x),f(y)) {\lt} d(x,y)\quad . \]

Show that \(f\) has a single fixed point.

This result is sometimes called Edelstein’s Theorem.

Hidden solution: [UNACCESSIBLE UUID ’27C’]

[UNACCESSIBLE UUID ’0TJ’]