Let ƒ : [0, 1] → R be a continuous function. We let ran(f) = {y | y = f(x) for some x = [0, 1] }. Prove that ran(f) is a closed interval. Prove that any non-empty closed interval is equal to ran(f) for some continuous function f : [0, 1] → R. Let f: [0, 1] → R be a continuous function such that f(0) = f(1). Prove that there is some x = [0, 1] such that f(x) = f(x+1).
Let ƒ : [0, 1] → R be a continuous function. We let ran(f) = {y | y = f(x) for some x = [0, 1] }. Prove that ran(f) is a closed interval. Prove that any non-empty closed interval is equal to ran(f) for some continuous function f : [0, 1] → R. Let f: [0, 1] → R be a continuous function such that f(0) = f(1). Prove that there is some x = [0, 1] such that f(x) = f(x+1).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![1. CONTINUOUS FUNCTIONS ON CLOSED INTERVALS
(a) Let f: [0, 1] →→ R be a continuous function. We let
ran(f) = {y|y=f(x) for some r = [0, 1] }.
Prove that ran(f) is a closed interval.
(b) Prove that any non-empty closed interval is equal to ran(f) for some continuous function
f: [0, 1] → R.
(c) Let f: [0, 1] → R be a continuous function such that f(0) = f(1). Prove that there is some
x = [0,] such that f(x) = f(x+).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F04625ac1-43ff-4999-b93a-55388fc0c5e2%2F7e82aadd-f6ec-4f9b-b718-3ca665379b96%2Fki8d3r2_processed.png&w=3840&q=75)
Transcribed Image Text:1. CONTINUOUS FUNCTIONS ON CLOSED INTERVALS
(a) Let f: [0, 1] →→ R be a continuous function. We let
ran(f) = {y|y=f(x) for some r = [0, 1] }.
Prove that ran(f) is a closed interval.
(b) Prove that any non-empty closed interval is equal to ran(f) for some continuous function
f: [0, 1] → R.
(c) Let f: [0, 1] → R be a continuous function such that f(0) = f(1). Prove that there is some
x = [0,] such that f(x) = f(x+).
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