A fundamental fact is that if we are working on a closed, bounded interval[a, b], then any continuous function ƒ : [a, b] → R is automatically uniformlycontinuous.Theorem. Suppose f [a, b] → R is continuous. Then it is uniformlycontinuous.8. Prove this theorem by arguing by contradiction, using the the previousquestion and the Bolzano-Weierstrass theorem.

Let I=a,b If possible let f be not uniformly continuous on I. Then there exist a positive ε0 for…

Answered: Theorem. Suppose ƒ : [a, b] → R is…

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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need 1 (a) (i) (ii) and (b)
1. Suppose f is continuous on R and f satisfies f(x) + f(2x) = 0 for all x E R. Prove that f = 0 on R.
1. Define a function f : Z x Z → Z by f(m, n) = 2n – 4m. (a) Prove that ƒ is not one-to-one. (b) Prove that the range of f is equal to the even numbers. In other words, show rng f = {2k|k E Z}, by using the usual methods for proving an equality of sets.
3. Is there a function f: R² → R that is not continuous but f(U) is open in R for all open sets UC R²?
4. Let f:A→R and g:B→R where f(A)CB. Explain to prove that if f is continuous at CE A and g is continuous at f (c)e B , then g of : A→R is continuous at c.
2. Let f : R → R be a continuous function. Suppose that (f o f) (x) = x for every x E R. (a) Prove that there exists a point c ER such that f (c) Hint: Consider the function g (x) = f (x) – x for every x € R. = C. (b) Give an example for a function satisfying the above requirements, and such that f id, -id , where id is the identity function.
Suppose f: n is a function and write f (fi,..., fm)" where fi: - R (assume the real line has the standard absolute value metric here). In other words, we write f(#) = (fi(7),..., fm(F))" . Then f is not continuous if at least one of the f i is not continuous. True False
!
3. Assume that f(x) is continuous on [a, b] and x₁,x2,...,n are arbitrary numbers taken from [a, b]. Show that there exists c E (a, b) so that f(c) 1 · [ƒ(x1) + ƒ(x2) + ... + f(xn)]. n == Hint. Use Intermediate and Extreme Value Theorems.
M 3. Let f, g be C1 functions on an open interval containing [a, b]and assume f ≤ g on [a, b]. DefineS ={(x, y)∈R2 : f (y)≤ x ≤ g(y), a ≤ y ≤ b}.(3a) Prove that S is Jordan measurable. Include a sketch to illustrate your formal argument.
1. Use the e-6 definition to prove that f(x) = r² is uniformly'continuous on [a, b).

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Theorem. Suppose ƒ : [a, b] → R is continuous. Then it is uniformly continuous. 8. Prove this theorem by arguing by contradiction, using the the previous question and the Bolzano-Weierstrass theorem.