Let f: V → II Xabe defined as: a € A f(v) = (fa(v)) a € A where fa: V → X₁ for each a € A. Let II Xa have the product fopology, then the function f is continuous if and only if each function fa is continuous. a € A

Q: Find an example of a function ƒ: N → Z which is (a) injective but not surjective (b) neither…

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Q: 5x+1 9. Prove that the function ƒ : R− {2} → R− {5} defined by f(x) = x-2 is bijective.

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A: For the solution of the problem follow the next steps.

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Q: Suppose f X→ Y, and A, B C X. Prove that (a) f(AUB) = ƒ(A) U ƒ(B) (b) ƒ(ANB) ≤ ƒ(A) Ñ ƒ(B)

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Q: et f: AB and g: B→C be functions. Show that if gof is njective, then ƒ is injective.

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Q: Exercise 1. Let f : A → B be a function with X a subset of A. Prove that if f(X) = 0, then X = 0.

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Q: a-c

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Q: Suppose and z are functions of x and y, then prove.

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Q: Let F(R, R) be the set of all functions from R to R. For any f, g € F(R, R), we define fog € F(R, R)…

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Q: If a function f: D R is differentiable at a point (a, b)ED, then f is continuous at (a, b);

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Q: Theorem 11. Let f be a continuous function on R such that f(x+y)-f(x)+fv) vx, yER. Show that f(x)3cx…

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Q: (b) ƒ(AnB) ≤ f(A) Ñ ƒ(B)

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Q: Metrics. Show that, for fixed x0 in E, the function  from E into ℝ+ is uniformly continuous.

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Q: Prove: There exist functions f, g, h in F such that g o f = h o f, but g is not equal to h.

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Q: f: (R, +) → (R*, .) where f(x) = 2*

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Q: (c) g: Z²Z2 by g(m, n) (m+n, m+ 2n)

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Q: Consider a function f:C C, defined by f(z) = 2z2 Then which of the following is true? Check all that…

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Q: Consider the function f : R →R defined by f (x) = 2x + 1. This function is:

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Q: g(z) = 1+1 and h(z) = [r²]. %3D Let I = (0, 1] and J = {1,2,3, 4}. A) What is (goh)(x)? B) What is…

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A: Given: f:X→Y and a,b⊆X, Find out: To choose the correct option.

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