16. Suppose B is a o-field of subsets of R. Show B(R) C Biff every real valued continuous function is measurable with respect to B and therefore B(R) is the smallest o-field with respect to which all the continuous functions are measurable.

Q: if x 4.

A: The function gx is a piecewise defined function.

Q: 7. (10 points) Let f: A B be onto. Prove that S((C)) = C for all sets CC B.

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Q: 7. Prove that the function f:Z → Z defined by f (x) = x + 8 is one-to-one and onto.

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Q: What does it mean to say f : (a,b) →R is a bounded function? If a < b in R and f : (a,b) → R is…

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Q: 7. Prove that if f and g are continuous then so are max(f, g) and min(ƒ, g).

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Q: Prove that there is an additive function T : R --&gt; R that is not linear.

A: Let T: V→W  be the linear transformation and we it is linear,…

Q: f1, f2, …, fn  each be complex functions defined on a metric space, X.  Suppose each of f1, f2, …,…

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Q: 2. Let N = {1,2,3,...}. Are the following functions injective, surjective and/or bijec- tive? (i) f:…

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Q: 5. Let f be defined for all real numbers and suppose f(x)-f(y)| ≤ (x - y)² for all real numbers, x…

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Q: Let A, B C X be subsets and f : X → Y a function. Show that ƒ(AÑ B) ≤ f(A)Ñ ƒ(B)

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Q: Show that f is continuous on (∞ , -∞ )  .

A: we have to show that f(X) is continuous in (-∞,∞). f(x)=x2             if   x&lt;1           x…

Q: Count the number X of functions y: (20, 21,..., 23) → (5,6,...,8) so that is not onto. X =

A: Given function φ:(20,21,...,23)→(5,6,...,8)

Q: 5. Let f be defined by the floor function f(x) = [] and let S = {-3, -2, -1, 0, 1, 2, 3}. a)…

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Q: 4. Let A C R" be both connected and sequentially compact. Prove that if F: A –→ R is continuous,…

A: Consider the given information.

Q: 5. Prove that if ƒ : [-1, 1] → R is measurable, then |f| is also measurable.

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Q: 6. A function f : R → R is said to be bounded if |f(x)| M, for some M > 0. Give an example of…

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Q: if f is measurable then - f is messlurable

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Q: Let A =[1 -1]            [2 0] and --->x =[ i]                  [1 + i] . SHOW that Re (A--->x )…

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Q: 6. Let f(x) = x55x+1. Prove that f has at least 3 zeros in [-2, 2]; that is, there exist distinct…

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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: Determine whether each of these functions from Z to Z is one-to-one. a) f(n) = 3n+3 b) f(n) = n²+1…

A: We have to determined whether this function from Z to Z is one to one. As per our guidelines we are…

Q: Let f(x, y) = domain of f is the set D = {(x, y)|x² + y² > 1}. Select one: 1 1 - x² - y² True O…

A: Given.

Q: 21. Let f(n) = [/n] – [/n – 1]. Prove that f is multiplicative but not completely multiplicative.

A: See the attachment

Q: 6. Let f: A B be bijective. Prove that f-¹ of = A. -

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Q: 9: Determine whether fis injective, surjective or bijective. TROPHE ZBAN DAN d. Suppose f: N→ N has…

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Q: 4. Let f : R+ x Z → R+ be the function defined by f (x,n) = x", where R+ is the set of positive real…

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Q: 20. Give two examples of functions from Z to Z that are one- to-one but not onto.

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Q: 4. `s.] Find the number a and b at which f is continuous everywhere? x² -1 if x 2

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Q: 11. Consider the function 9: {0, 1}xN-Z defined as 8(a, b)=(-1)ºb. Is 8 injective? Is it surjective?…

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Q: Let A= R- {o,}. det G = {4, t, qf where 1: A> A defined by icx)> x z¢A L+ ズ-A fex) = 1ーメ+x-A 1-X 9:A…

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Q: 1.49. (!) Let f and g be functions from R to R. For the sum and product of f and g (see Definition…

A: