4. Suppose that f,9: [0, 1] - (0, ox] are continuous functions and thatsup{f(x) : E (0, 1]} = sup{g(r): r€ [0,1]}.Prove that there exists t € (0, 1] such that f(t) = g(t).5. Given that f: [0, 1] - R satisfiesIf(r) – f(y)| < |x - ylfor all r, y e [0, 1] with z # y. Can We conclude that there exists a constant K <1 such that for alla, y € [0, 1]If(2) – f(y)| < K|a - y?Hint: consider f(r) = sin r.

As per the guideline, I would only do the number 5. Kindly repost the other question separately.…

5. Given that f: [0, 1] →R satisfies If(z) - f(u)| < |x - yl for all r, y € (0, 1] with z#y. Can We conclude that there exists a constant K <1 such that for all z,y € (0, 1] IS(2) - f(y) S Kz- yl? Hint: consider f(r) = sin r.

5. Given that f: [0, 1] →R satisfies If(z) - f(u)| < |x - yl for all r, y € (0, 1] with z#y. Can We conclude that there exists a constant K <1 such that for all z,y € (0, 1] IS(2) - f(y) S Kz- yl? Hint: consider f(r) = sin r.

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter6: Vector Spaces

Section6.5: The Kernel And Range Of A Linear Transformation

Problem 30EQ

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Question

4. Suppose that f,9: [0, 1] - (0, ox] are continuous functions and that
sup{f(x) : E (0, 1]} = sup{g(r): r€ [0,1]}.
Prove that there exists t € (0, 1] such that f(t) = g(t).
5. Given that f: [0, 1] - R satisfies
If(r) – f(y)| < |x - yl
for all r, y e [0, 1] with z # y. Can We conclude that there exists a constant K <1 such that for all
a, y € [0, 1]
If(2) – f(y)| < K|a - y?
Hint: consider f(r) = sin r.

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