Homework 6

2. [Alternative Proof of Intermediate Value Theorem]
(a) Prove the following proposition:
Let f be a real-valued function defined on an interval S in R. Assume that f is
continuous at a point c in S and that f (c) ≠ 0. Then there is an open ball B δ (c)
such that f(x) has the same sign as f(c) in B δ (c) ∩ S.
(b) Prove the following theorem:
Let f be a real-valued and continuous function on a compact interval [α, β] in R,
and suppose that f(α) and f(β) have opposite signs. Then there is at least one
point γ in the open interval (α, β) such that f(γ) = 0.
- Hints: For definiteness, assume f(α) > 0 and f(β) < 0. Define the set
A = {x: x ∈ [α, β] and f(x) ≥ 0} .
Then A is nonempty since α ∈ A, and A is bounded above by β. Let γ =
supremum of A. Then α < γ < β. Prove that f(γ) = 0. [Suppose not, and
then use (a) to come up with a contradiction.]
(c) Use (b) to prove the following version of the intermediate value theorem:
Let f be a real-valued continuous function defined on an interval containing the
real numbers a and b, with say f(a) < f(b). Then, given any y ∈ R such that
f(a) < y < f(b), there exists x ∈ (a, b) such that f(x) = y.
3. Let f : [a, b] → [a, b] be a continuous function on [a, b]. Prove that there exists some
β ∈ [a, b] such that f (β) = β.
4. Let S be a closed interval and let 0 < α < 1. Let f : S → S satisfies the inequality
|f (x) − f (y)| ≤ α |x − y| , for each x ∈ S and y ∈ S.
(a) Prove that f is continuous on S.
(b) Let x 1 ∈ S and define a sequence of real numbers whose (n + 1)th element is
x n+1 = f (x n ) , n = 1, 2, 3, ... Prove that this sequence is a Cauchy sequence.
[Hint: |x n+1 − x n | = |f (x n ) − f (x n−1 )| ≤ α |x n − x n−1 | .]
(c) Use (a) and (b) to prove that there exists x ∗ ∈ S such that f (x ∗ ) = x ∗ .
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