Q14 Let X= R. Define a function : P(R) → [0,0] by #* (A) = { #A (a) Show that is an outer measure on R. #Aif A is finite, if A is infinite. (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 Borelo-algebra on R

Q: For S in the domain of f, let f ( 5 ) = { f(x): * € S}. Let C and D f de subsets of the domain of f.…

A: S is in the domain of f fS=fx: x∈S Let C and D be subsets of the domain of f Then we have to show…

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: 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: Prove that if f:X + Y is a one-to-one and onto function with inverse function f- : Y → X, then fof-l…

A: Given that, f:X→Y is a one-to-one and onto function with inverse function f-1:Y→X

Q: Let function f: Z Z be defined as follows; 1 x is odd f(x) = . x² + 2 x is even The functionf is .…

A: To show that  a function f(x) is one-one , we have to show that for all x , y belong to the given…

Q: Let ƒ be continuous and nonnegative on [a, b]. Show that there is c € [a, b] with f(c) = (±a få…

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Q: 11. with f(-1)=6, f(1) = 8 and f(6) = 3. Intermediate Value Theorem? Let f be a function that is…

A: Given:f be a function that is differentiable on the closed interval -1, 6 such…

Q: Let f : R → R be defined by f(x) = x^2. Is f one-to-one? Explain your answer.  Is f onto? Explain…

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Q: Let h : R → R be some twice-differentiable function. Define f(x,y) = h(x) for all x ≠ 0. Show that…

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Q: 4. Let X an Y be sets with A, B C X and let f: X→Y be a function. Recall that {f(x) EYE A}, and…

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Q: f : R → R+ is a function such that f(x y) = f(x) − f(y) for all x,y ∈ R+. Prove thefollowing (a) f…

A: Let the given function is defined as f (x + y) = f (x) – f (y). find the continuity of the function…

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: Consider the function f : [−2, 6] → R, x→ f(x) = 2x3/ [√(x2+8)+2]. i) provide argument why this…

A: This is the question of Real topology, and real analysis.

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: 3. Let z € R" be denoted by x = (₁, 2,...,n) and || x ||:= √√x² + x² + +x². Let f: RR be defined by…

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Q: (b) Let X = (a, b]. Construct a continuous function f : X → R (set of real numbers) which is…

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Q: Let f:Z x Z+ Z x Z be the function given by f(z, y)= (2z + y,I- y). Find, with proof, an element af…

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Q: Let I be an interval and f : I → R be a function with Darboux property such that f(I) is finite.…

A: Let I be an interval and f : I → R be a function with Darboux propertysuch that f(I) is finite.…

Q: Let a function f be analytic everywhere in a domain D. Suppose that f(z) is pure imaginary for all z…

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Q: ) Let f be a continuous function that maps a finite closed interval [a, b] into itself. Prove that f…

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Q: Let X, Y and Z be sets. A, B are subsets of Y. f:X → Y is a function. Prove the given statement if…

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Q: Let f be differentiable on R. Suppose that f'(r) # 0 for every r. Prove that f has at most one real…

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

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Q: If f,g continuous on [a,b] and f(a) g(b). Show that there exists c ∈ (a,b) such that f(c) = g(c)

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Q: Prove that f is contraction map if and only if (X,τd)  to ( Y,τc)  is continuous

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