91Ó°ÊÓ

Give examples of functions which do not have fixed points but do have the following characteristics: (i) \(f:[0,1] \rightarrow[0,1]\) (ii) \(f:(0,1) \rightarrow(0,1)\) and is continuous (iii) \(f: A \rightarrow A\) and is continuous, with \(A=[0,1] \cup[2,3]\) (iv) \(f: \mathbb{R} \rightarrow \mathbb{R}\) and is continuous.

Show that \(f(x)=-x / 2\) is a contraction on \([-2,-1] \cup[1,2]\) but has no fixed point there. Why does this not contradict \(\mathbf{7 . 2} ?\)