Proposition 7.2. Interchanging integral and limit Suppose fn E R[a, b] and fn → f uniformly on [a, b]. Then f E R[a, b] and . fn → S, f.

Q: 1 Let f (x,y) =In Then f (1, – 1) =... 3x2+4y3 XX

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Q: 1. Let f: Z/4Z → Z/2Z × Z/2Z be defined by f([0]) = ([0], [0]), ƒ([1]) = ([1], [1]), f([2]) = ([0],…

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Q: Q. Let g(x) : [2, 3] -+ R be a continuous mapping. 3 And / (9(x) + 2 – 1)'dx = 0, then - a.g(x) = 0.…

A: Given function g(x) is continuous on the interval [2,3]

Q: PROPOSITION 7.17. Suppose that f, g: [a, b] → R are integrable. Then: (1) For any k € R, k. f is…

A: Given: 7.17) f, ga, b→ℝ are integrable. To show: 1) For any k∈ℝ, k·f is integrable on a, b and…

Q: For the given matrix: 2 -1 2 -1 0 -1 0 -1 2 -1 a) Compute the determinant. b) Calculate the rank.

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Q: State TRUE or FALSE giving proper justification for the following statement. There exists a…

A: The given statement is : "There exists a continuous function f:ℝ→ℝ2 such that f(cosn)=(n,1n) for all…

Q: Let f: R - R be defined by f(x) = |x+2, r20 interval ]1, 3[ under f) is (a) ]3, 5[U] – 1, 1[ (b) (0.…

A: given that,  f(x)=x-2, x<0x+2, x≥0

Q: True or False? If f : (0,1] → R and g : [1,2) → R are uniformly continuous on their domains, and…

A: True

Q: Suppose f: C → C is a holomorphic function. Let g(z) = ƒ(₹). Using the Cauchy Riemann equations, or…

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Q: 4. Let f(z) = cosh z. (3) Show that f: S→ f(S) is ono-to-one.

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Q: f(z) = Im z is continuous 0 صواب ihi O

A: True

Q: 1.3 Let fn: (-3, 3] R be defined by fa(r) Find the pointwise limit of fa(x). %3D

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Q: Theorem (6) If f is bounded and R – I on [a, b], f(x) > 0, Vx € [a, b] then ſ f 2 0

A: Prove that: If f is bounded and R-I on [a, b], fx≥0 for all x∈a, b , then ∫f≥0,

Q: (4) Suppose f(x) is increasing on [a, b]. Let Pn = {0, 1,...,n}, where xk = a + k (b − a), k = 0,…

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Q: Let E = {0, 2, 4, 6, ...}, define f: Ex E → Z+ by the formula f(m, n) = 4m5¹, v (m, n) € E × E. (a).…

A: Let Ɛ = {0, 2, 4, 6,...}, define f: Ɛ x Ɛ → Z+ by the formula f(m, n) = 4m5n for all (m, n) E Ɛ x…

Q: Bölüm 3 ) Let R= (a,b] x [c, d]. Let f: R R function be defined as f (x ,y) = gx) hly) hi [c,d] →R…

A: Here I am attaching image so that you understand each and every step.

Q: Let f be a continuous function on [0, 1] with f(0) = f(1). (a) By considering g(x) = f(x) f(x + ¹)…

A: Sol:- (a) Since f(0) = f(1), we have g(0) = f(0) - f12 = f(1) - f12 = -g12. Since g is continuous on…

Q: 3. Let fn(x) = ¹+cos(nx), x € R. Find f(x) so that fn → ƒ pointwise on R, then check whether fn → f…

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Q: 5. Prove that |xy| = |x|-|y| for x, y eR.

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Q: Suppose that f(x, y, z) = (1+x + y² + z³)4 + ln (1 + 3x + 2y + z).

A: In this question, we approximate the function by using a linear approximation of f(x,y,z)

Q: 6. Let fn (n = 1, 2, 3, ...) be respectively: (a) n² X[1 1 (b) (c) X[n] 1 3kX[k,k+1] n k=0

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Q: a.s. Let X, X, E|X,| < x, E|X| < x, and EX, → EX. Prove that E|X, - X| → 0. (Hint: dominate the…

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Q: 2. Express the following limits as a definite integral of a suitable function over a finite…

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Q: → Det f : V → Il Xa be defined as : a e A f (v) = (fa(v)) a e A where fa : V ¸Xa for each a e A. Let…

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Q: Let f:R → R given by f(x) = 2x² and A = [-3, 1] U [2, 4]. Then f(A) is

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Q: 1. Show that if f: [a,b] → R and a < xo <b, then f'(xo) exists if and only if D f(x) = Dƒ(x) € R.

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Q: 14 Jlgull s(e)= f(2) =Re z is continuous ylgn O Ihs O

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Q: 1. Give an example of a function f : R –→ R that is continuous and bounded but does not achieve…

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: Suppose f: -1, 1] → R is continuous and satisfies f(-1) = f(1). Prove that there exists 7 € [0, 1]…

A: Given is a continuous function.

Q: 2. Let function f [0, 1] → R defined by f(x) = integrable on [0, 1]. Evaluate f f(x) dr. 659 0, if a…

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Q: 1. Let f(x) = x. Show that f is uniformly continuous on any finite closed interval [a, B] (Hint: x-y…

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Q: 4. (a) Let I = [a, b] and let f and g be bounded functions on I to R. Suppose that f(x) < g(x) for…

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