Q1 (4 Let XR. Define a function : P(R)→→ [0,0] by μ*(A) = (a) Show that is an outer measure on R. #Aif A is finite, if A is infinite. 8 (b) Show that every ACR is -measurable and hence deduce that in fact defines a measure on P(R). (c) Prove that every function f: R→ Ris (P(R), B(R))-measurable, where B(R) denotes the Borel o-algebra on R (d) Let f: R→ [0, 0o] be such that the set E= {x Rf(z) 0} is infinite and countable. Writing E = {x}₁. prove that [ƒdu" = f(x). 3-1 t: Choose a good sequence of simple functions f such that ff pointwise. 1-

Q: Let f be a function from N to (0,1). Prove -h for any nEN, y = f(n). Using part (a), prove that…

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Q: Define f : R2 → R by letting (y² – x)² (y4 + x²) f(x, y) = if (x, y) # (0, 0), and f(0,0) = 1. ||…

A: The given function is f:ℝ2→ℝ defined as: fx,y=y2-x2y4+x2if x,y≠0,01if x,y=0,0 Claim: The given…

Q: Let (X, E, µ) be a measure space and let ø, v be nonnegative simple functions. Show that |(6+ v)dµ =…

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Q: b) Give an example to show that the product fg need not be uniformly continuous.

A: Given: f:A→B, g:B→C are uniformly continuous.To show: b) The product fg need not be uniformly…

Q: P 2. ( a) Find the average value of f on [9,25]. b) Find all c in [9, 25] such that f(c) = favg.…

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Q: Let F be a continuous function on [a, c} , where a < b< e, and F(z) dz = 6, F(2) dz = 9 A. Evaluate…

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Q: Let f: R (-1, 1) be a function defined by Prove that (a) f is injective; (b) f is surjective. f(x) =…

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Q: 2. Let f be defined on an open interval (a, b) and assume that for some e e (a, b) we have f'(c) > 0…

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Q: → R be a function defined by 3x² + x+5 f(x) = not continuous at x = = 0.

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Q: "Prove that the function f : R→ R with ƒ(x) = = is uniformly continuous."

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Q: Exercise 4: Suppose f is differentiable on (2,6) and continuous on [2,6. Suppose further that 1 < f'…

A: Given:f is differentiable on (2, 6) and continuous on [2, 6].

Q: Consider f e C} (R") and the following function 1 Hy (æ, z) = f(z) + Vf(z)"(x – z) + ||x – z||3, 27…

A: note : As per our company guidelines we are supposed to answer ?️only the first question. Kindly…

Q: Theorem 12. Let f be a non-zero continuous function on R such that f(x+y)-f(x) f(y) ♥x, yER. Show…

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Q: Ja - 2, x < 0, a+ 2, x20 9. Let f: R R be defined by f (x) = then f-(]1, 3() (the inverse image of…

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Q: F is a function from A to B. F is a function from P(A) --> P(B). If f is an injection prove F is an…

A: Given that f is a function from A to B that is f:A→B and F is function from PA to PB that is…

Q: Let f be a real-valued function on [a,b] that is differentiable on (a,b). Suppose f '(x) < 0 for all…

A: This is a problem of the application of differential calculus.

Q: 2: If f(x) and g(x) are continuous on (a,b]and if f =[s . Prove that 3cela,b] such that f(c)=g(c).…

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Q: Let ƒ : R → R be a function that is differentiable on (a, ∞), where a is any real onstant. Let g: RR…

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Q: example of a function f:[0,1]→ℝ such that f is not continuous, but f does satisfy the conclusions of…

A: Given function f:01→R f is not continuous, but f satisfies the conclusion of extreme value theorem

Q: f Let f:(R,1,) - (R,T,), f(x) = x². Find f-(a, b) and then show that is continuous on R.

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Q: f(x)=xmodx for all x  in R. Discuss the derivability of f(x) at x=0

A: Given:fx=xxwe have to find derivability of fx at x=0

Q: If that f is continuous on a sxsb. Let F(x) be any of f(x). Then, f(x)dx = F(b) - F(a). This theorem…

A: Recall: Mean value theorem: If f(x) is continuous over an interval [a,b], then there is atleast one…

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: Define f: Z x Z→ Z x Z by f((m, n)) = (3, 2n). Let A = {(n, 0): n ∈Z}. Fina f(A).

A: EXPLANATION : Given function  f:ℤ×ℤ→ℤ×ℤ defined by fm,n=(3,2n). given set A={(n,0):n∈ℤ}. putting the…

Q: A function is said to be Hoelder a continuous on [a, b] iff |f(x) – f(x')| 0. Show that if a > 1…

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Q: If fCxl is continuous on [2,7],then Si f(u) dvz f (7)-(2) True e or false?

A: we have given that f(x) is continuous on[2,7].

Q: Let A := (0, 1] and let ƒ : A → R be defined by f(x) = !. Prove that ƒ is continuous on A.

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Q: Consider a function ƒ : [0, 1] → R defined by f(x) = x(x³ - 1). (a) Can we apply Rolle's Theorem to…

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