Analysis II | Part IB, 2001

Let ff be a mapping of a metric space (X,d)(X, d) into itself such that d(f(x),f(y))<d(f(x), f(y))< d(x,y)d(x, y) for all distinct x,yx, y in X\mathrm{X}.

Show that f(x)f(x) and d(x,f(x))d(x, f(x)) are continuous functions of xx.

Now suppose that (X,d)(X, d) is compact and let

h=infxXd(x,f(x))h=\inf _{x \in X} d(x, f(x))

Show that we cannot have h>0h>0.

[You may assume that a continuous function on a compact metric space is bounded and attains its bounds.]

Deduce that ff possesses a fixed point, and that it is unique.

Typos? Please submit corrections to this page on GitHub.