Just (c) from the problem below:
1.11. Let f be a one-to-one smooth map of the real lineto itself. One-to-one means that if f(xi) = f(x2), then x1 = x2. Afunction f is called increasing if x1 ? x2 implies
f(x1 )?f(x2), and decreasing if x1 ? x2 implies f(x1 ) ?f(x2 )
(a) Show that f is increasing for all x or f isdecreasing for all:x |
(b)Show that every orbit {x0, x1, x2…} of f2 satisfieseither 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 orbitsof f? |
