(Term Project)[73] Let f : I → I be a one-to-one continuous map on the closed bounded interval I. (a) Show that either f is strictly increasing or f is strictly decreasing on I. (b) Show that f is not...

(Term Project)[73] Let f : I → I be a one-to-one continuous map on the closed bounded interval I.

(a) Show that either f is strictly increasing or f is strictly decreasing on I.

(b) Show that f is not transitive on I. (c) Show that if f is strictly increasing, then every periodic point is a fixed point. Moreover, if f has a non-fixed point, then the set of periodic points Perf is not dense in I.

(d) Show that if f striclty decreasing, then there is exactly one fixed point and all other periodic points are of period 2. Moreover, if
 for some x ∈ I, then the set of periodic points Perf is not dense in I.