4. Let f be defined for all real x, and suppose that \fx) – f(v)| < (x – y)? for all x, y e R. Prove that f is constant.

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: 31. Let f(x) = [x²/3]. Find f(S) if a) S= (-2,-1, 0, 1, 2, 3). b) S = {0, 1, 2, 3, 4, 5). c) S = {1,…

A: The given function is, fx=⌊x2/3⌋. To Find: f(S) for the following set S, (a) S=-2,-1,0,1,2,3 (b)…

Q: Let f(z) be an entire function satisfying that f(z)| ≤|z² for all z. Show that f(z) = az² for some…

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Q: Suppose that f(x) is convex and continuous on an interval I. Show that dif(x1) + ·.+ A„f(xn) + Anf…

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Q: f Gx,y) x+ 3 XY %3D

A: We have to check

Q: Let f(x, y) = (2x - y,x-2y); (x, y) E R x R, (R is set of real numbers.) %3D a) Show that f is one…

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Q: 27. Let f(x)= [x²/3]. Find /(S) if a) S-2, 1, 0, 1, 2, 3). b) S= (0, 1, 2, 3, 4, 5). c) S = (1,5. 7.…

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Q: 4. Let A and B be sets and let T C B. Also let f : A → B. (a) Use a diagram to show that f(f-'(T)) C…

A: According to the given information,

Q: If y =f (x) is a function in x then prove that f(a + nh) ="C,f(a) +"C, Aƒ(a) +"C,ƒ(@) + +"C, A"…

A: Given: y=fx is a function in x Prove that: f(a+nh)=C0nfa+C1n∆fa+C2n∆2fa+…+Cnn∆nfa…

Q: Find the values of a and b that make f continuous everywhere. b = f(x): x-2 ax² - 6x + 3 4x -a + b…

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Q: 1. Describe all functions f such that f(r) ar² + bx +c and f(1) = 2 and f(-1) = 6.

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Q: Consider the following statements 1- The function f defined by f(x) = x³ is decreasing in R 2- If a…

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Q: 3. If f:(X,r)(Y.A) and g:(Y,A)→(Z,r) are open functions, prove that gof :(x.r)(Z.r) is open…

A: Definition: An open map is a function between two topological spaces that maps open sets to open…

Q: Let f(2) = z2 + 2|2|2 + z where f is differentiable. Determine the set of all points zo where f is…

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Q: 6. Prove that |xy| = |x|ly| for all real numbers x, y.

A: The given problem is to prove the following statement. Given statement is |xy|=|x||y| for all real…

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

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Q: Z is the set of integers, R is the set of real numbers. 3) Each function f is defined as f: R -> R.…

A: The objective is to find the function is one-one or not one-one. (a) fx=x+1x-1for all real numbers x…

Q: Let f be a differentiable real function defined in (a, b). Prove that f is convex if and only if f'…

A: A function f is convex in a domain a,b if for all 0≤λ≤1, fλx1+(1-λ)x2≤λfx1+1-λfx2  ∀x1,x2∈a,b. Also…

Q: b) Is f one-to-one? Answer yes or no and explain. Do not just quote the definition, but give a…

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Q: 호호 -^ z-vz44 Z

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Q: Let f : A → B be a function. Prove that if C C A and D C A and f is injective, then f(C) = 7 (D) =→…

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Q: Select the following functions that are bijective. ƒ₂: R → R, ƒ₂(x) = 14 f₂: R → R, f₂(x) = x f₁: R…

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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: c) Is f onto? Answer yes or no and explain. Do not just quote the definition, but give a reason that…

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Q: 4. Show that the function f: R → R defined by f(x) = x² - 3x + 1 is not one-to-one.

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Q: A continuous real-valued function f (x) with f (a) ƒ (b) < 0 , has at least one root in [a, b]

A: This question is related to Real Analysis.

Q: (b) State the range of f for the domain given in part (a). f:A B f(x)=(x-1)²+4. (c) Define the set B…

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Q: Suppose f is a differentiable function on an interval II and f(x) 6= 0 for every x in the interval.…

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Q: a) Find the image set f (S) for S = {-2, 5}.

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Q: Is the set X of all functions f: Z+→ {0, 1} countable? Prove or disprove.

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Q: Suppose that f(x) is convex on an interval I. Show that f(x1) + f(x2) + ·+ f(xn) n for arbitrary n…

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Q: Let S(x²+2x)², x> 1, f(x) = Tax+ bx², ах x< 1. Compute for all a and b such that the function f(x)…

A: Proceed as shown

Q: Let f(x,y)=g(x)h(y) where g and h are continuous functions on all real numbers such that g(x)=h(x+1)…

A: