Example where Banach fix point theorem isn't valid

Question

Can someone give me an example of a complete metric space that doesn't have fix point theorem? (Counterexample of fixed point Banach theorem).

Answer 1

Classic example of function with $d(f(x),f(y))<d(x,y)$ and without fixed point in a complete space $$f(x) = x + \frac1x,\qquad X = [1,\infty).$$