11. Let f be integrable over E where E = AUB is a disjoint union of measurable sets. Prove that [-]+[s. f+ E B

Q: 'he function f: E → R is uniformly continuous if

A: We have to find the answer using the definition of uniform continuity.

Q: 2. Show that if f is continuous on [a, b] with f(x) > 0 for all a ≤ x ≤ b, then 1/f is bounded on…

A:

Q: Define the function fn : [0, 1] –→R for n > 1, defined as if æ E (,1] nQ if x E (, 1]\Q if x E [0,…

A:

Q: 7. Let u be transcendental over F and F< EC F(u). Show that u is algebraic over E.

A: Algebraic field

Q: 3. Let f : [2,5] → R such that f(x) et Is f uniformly continuous? Justify your answer. Answer:

A:

Q: Theorem 4. Let X be a set, and let F: X Y be a l-1 and onto function. (a) F-lo F = Idx (b) FoF- =…

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Q: Let S=X belongs to R: x> =1. a) Show that the set S has lower bound, but no upper bounds. b) Show…

A: The answer is given as follows:

Q: Let f : A → B be a function, {B;}jej a family of subsets of B. Prove that f-'(Njej B;) = Njej…

A: Given that f : A → B is a function. Also given that Bjj∈J is a family of subsets of B . We have to…

Q: suppose cX) is valued function on such that f?E R Faib] Does it fallows bounded real taib] that…

A: In the given question given that f(x) is a bounded real valued function on [a,b] then we have to…

Q: 10) Prove that if fεg are continuous functions R on IR then fog is also continuous on i

A: Step 1:To prove that if f and g are continuous functions on R then fog  is also continuous on R, we…

Q: 1. Let f: RR, f(x) = x³. a. Show that f is continuous on R b. Show that f is not uniformly…

A:

Q: [0,1] 6. If f is continuous on • prove that

A: We can solve an integral of some function f(x) w.r.t. x by using substituting, say, t=f(x) which…

Q: 2.58 Given an example of f E C (a, b) for which the range of f is a) a finite open interval. b) an…

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Q: Which of the following function f: Z x Z Z is not onto? 2020 a. f(a, b) = a b. f(a, b) = lbl c. f(a,…

A:

Q: Provide two disjoint nonempty open sets C, D. Is it possible for a function g : R → R to be…

A: A set is a collection of elements like, numbers, objects. The numbers can be called as set of real…

Q: Prove: If f1, f2 € R(a) on [a, b] with fi < f2, then ' fi da < s' fz da

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Q: Sj. Let f be a measurable function with domain DER and let ECD be a measurable subset of D. Show…

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Q: Let A be a non-empty set of positive numbers, which is bounded above and let B = {x e R: 1/x e A}.…

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Q: 2. Let a € R be given. Define f: R" → R by f(x) = a¹x, x = R¹. Using the € - & definition, show that…

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Q: 8. Prove that a function which is uniformly continuous on X is also continuous on X

A:

Q: et F be the collection of

A: To define, T(G)=∫G(z-y)dG(y),G∈F

Q: compute measuralle

A: [f&gt;t] is an inverse image of (t,infinity) That leads us to give an answer.

Q: 3. Give an example of a non-Lebesgue measurable function f: IR→ IR such that || is a measurable…

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Q: If f is continuous and periodic on R, prove that f uniformly contiuous on R.

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Q: Exercise 1.4.4 (Hard): Construct an explicit bijection from (0, 1] to (0, 1). Hint: One approach is…

A: In this solution we will find an explicit bijection from (0,1] to 0,1. We know that a function f:A→B…

Q: Show that if f and g are continuous on [a, b] and if f(a) g (b), then there exists some ce ]a, b[…

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Q: If f, g E R[a,b] with f < g on [a,b], prove that g.

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Q: If f, g: R → R s.t. f and g are uniformly continuous. and bounded, then show that f.g is also…

A: Let f ,g are uniformly continuous And f and g are bounded

Q: Let (X, E, µ) be a measure space and let {fn} be a sequence of nonnegative measurable functions on…

A:

Q: Suppose f,g: D -> R are uniformly continuous on D. Prove that f+g is uniformly continuous on D.

A: To Prove- Suppose f, g : D → ℝ are uniformly ccontinuous on D. Prove that f + g is uniformly…

Q: - Let A = {1, 2, 3, .., 103. Consider f:P(A) → N given by f(B) = |B|. That is, f takes B subset of A…

A:

Q: Let the function ƒ : [0, 1] →R be defined by { if x = ! (n = 1, 2, 3, · · · ) f(x) = n 1– x2…

A: Given :  f : [0, 1] →ℝ  defined by ;              fx = ex2,if  x=1n    (n=1, 2, 3, ...…

Q: Suppose f : N→N and g :N→N , function f is not onto but f+g is onto function  does g onto function…

A: g is not necessary to be onto. See the attachment

Q: Suppose that f : R → [0, 0) is measurable. (positive) show that if f R then F=0 a.e

A: For the solution follow the next step.