A fixed-point of a function f : A → A is a point a ∈ A such that f(a) = a. The diagonal of A × A is the set of all pairs (a, a) in A × A. (a) Show that f : A → A has a fixed-point if and only if the graph of f intersects the diagonal. (b) Prove that every continuous function f : [0, 1] → [0, 1] has at least one fixed-point. (c) Is the same true for continuous functions f : (0, 1) → (0, 1)?† (d) Is the same true for discontinuous functions?

Suppose that f : [a, b] → R is a continuous function that satisfies f (x) ∈ [a, b]for all x ∈ [a, b]. Prove that if f is differentiable on (a, b) and |f ′(x)| < 1 for all x ∈ (a, b),then f has exactly one fixed point. (Hint: We know that there is at least one fixed point byHomework 6. Try a proof by contradiction, and use the Mean Value Theorem.

the values of a and b, if the graphs of f (x) and )(1 xf − coincide.

Find an explicit bijection (i.e. give a formula that explains what f(x) is given any x ∈ [0, 1]) between [0, 1] and [0, 1).

QUESTION 26Choice D is correct. The definition of the graph of a function f in thexy-plane is the set of all points (x, f(x)). Thus, for −4 ≤ a ≤ 4, the valueof f(a) is 1 if and only if the unique point on the graph of f with x-coordinate a has y-coordinate equal to 1. The points on the graph of fwith x-coordinates −4, , and 3 are, respectively, (−4, 1), , and(3, 1). Therefore, all of the values of f given in I, II, and III are equalto 1.Choices A, B, and C are incorrect because they each omit at least onevalue of x for which f(x) = 1

Problem 2. For each of the following formulas for f (x), find the largest possible set A ⊆ Rso that f : A → R is a function, and prove that f is differentiable at every point a ∈ A.(a) f (x) = sec x = 1cos x .(b) f (x) = csc x = 1sin x .(c) f (x) = tan x = sin xcos x .(d) f (x) = cot x = cos xsin x