Question

Just (a) from the problem below:

1.11. Let f be a one-to-one smooth map of the real line to itself. One-to-one means that if f(xi) = f(x2), then x1 = x2. A function f is called increasing if x1 〈 x2 implies

f(x1 )くf(x2), and decreasing if x1 〈 x2 implies f(x1 ) 〉 f(x2 )

(a) Show that f is increasing for all x or f is decreasing for all:x

(b)Show that every orbit {x0, x1, x2…} of f2 satisfies either xo

x1x2… or xox1x2...

(c) Show that every orbit of f2 either diverges to +

or- or converges to a

fixed point of f2.

(d) What does this imply about convergence of the orbits of f?

Answer 1

Hence f is strictly increasing or strictly dicreasing on R.