An Arc as the inverse limit of a single bonding map of type N on [0,1]

Let I = [0, 1], c(1), c(2) is an element of (0, 1) with c(1) < c(2) and f : I --> I be a continuous map satisfying: f \([0, c1]) and f\([c2, 1]) are both strictly increasing and f\([c1, c2]) is strictly decreasing. Let A = {x is an element of [0, c(1)]\f(x) = x}, a = maxA, a(1) = max(A\{a}), and B = {x is an element of [c(2), 1]\f(x) = x}, b=minB, b(1) =min(B\{b}). Then the inverse limit (1, f) is an arc if and only if one of the following three conditions holds: (1) If c(1) < f(c(1)) <= c(2) (resp. c(1) <= f(c(2)) < c(2)), then f has a single fixed point, a period two orbit, but no points of period greater than two or f has more than one fixed point but no points of other periods, furthermore, if A not equal 0 and B not equal 0, then f(c(2)) > a (resp. f(c(1)) < b). (2) If f(c(1)) <= c(1) (resp. (c(2)) >= c(2)), then f has more than one fixed point, furthermore, if B not equal phi and A\{a} not equal phi, f(c(2)) >= a or if a(1) < f(c(2)) < a, f(2)(c(2)) > f(c(2)), (resp. f has more than one fixed point, furthermore, if A not equal 0 and B\{b} not equal phi, f(c(1)) less than or equal to b or if b < f(c(2)) < b(1), f(2)(c(1)) < f(c(1))). (3) If f(c(1)) > c(2) and f(c(2)) < c(1), then f has a single fixed point, a single period two orbit lying in I\(u, v) but no points of period greater than two, where u, v is an element of [c(1), c(2)] such that f(u) = c(2) and f(v) = c(1).