(a) Use the definition to prove that f(x) = is uniformly continuous on the x +1 domain [0, 00). x- 2 (b) Use the definition to prove that the function f(x): is continuous at a = 2. x+2

Q: Determine the domain of the function and prove that it is continuous on its domain using Theorems…

A: Given: f(x) = ln (9 - x 2) To determine: The domain of the function and prove that it is continuous…

Q: 1. Supposef is a continuous function on x 2 0,f'exists for x> 0, f(0) = 0 and f'is monotonically…

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Q: Find an example of a function f:[0,1]¬R such that f is not continuous, but f does satisfy the…

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Q: 2. Let f(x) = 2x²-3x+1 ,x 0.5 Determine constants A and B such that is continuous for all values of…

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Q: Find the value of the constant k so that f(x) is continuous for all real values of x where f(x) is…

A: Solution

Q: 3. Find two real-valued functions f and g that are uniformly continuous on a set E, but such that…

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Q: B) Let f:[-2,2] R, then show that the function sin for x 0, f(x) = 0) for x 0, is uniformly…

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Q: 6. Use definition of continuity to determine the value(s) of k such that the 2х + 5, х< 3 function f…

A: Topic : Continuity

Q: Let f(x)=x² -x² +x + 1 and g(x)= 3-х max{f(t):0sx; for 0 s x s 1} for 1 < x s 2 Discuss the…

A: Topic:- Functions

Q: Prove that the function f(x) = 2x is uniformly continuous on R.

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Q: f(x)={x^2−9x≤c, 2x−10 x>c } i

A: Consider function f(x)=x2-9           x≤c2x-10        x&gt;c Now  limx→c+ f(x)=limx→c+…

Q: Show that g(x) = (x3-8) / (x-2) has a removable discontinuity at x = 2. How should g(2) be defined…

A: The left-hand limit of a function at a point is the value of the function at a point very close from…

Q: 2. Let f(x) = S dt for x 2 2 In a) Find F'(x). b) Is F increasing or decreasing? What can you say…

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Q: a continuous function f such that f(4) = 4, f'(4) = 0, ƒ'"(x) 4 is:

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Q: c. Show that f(x) is NOT uniformly continuous on [0, +∞).

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Q: 8. Consider the function f(x)=/25-x² . a. Find the linearization L(x) of f(x) at x=3.

A: Given expression is , fx=25-x2 to find out the linearization L(x) of f(x) at x=3.

Q: Prove that f(x) =√x is uniformly continuous on [0,∞).

A: According to the question, we have to prove that the function fx=x is uniformly continuous on [0,∞).…

Q: Suppose that f is a continuous function for which f(1)÷0 and f(t) dt=0. What can we conclude about…

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Q: [0, 4). 4. Prove that f(x) = x² is both uniformly continuous and continuous on

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Q: 1. Suppose that f : [0, 0) → R is continuous and f(x) # 0 for all x > 0. If we have (f(x))? = 2…

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Q: B) Let f:[-2,2]R. then show that the function xsin -) for x 0, f(x) = for x = 0, is uniformly…

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Q: Assume that g : R → R is a periodic continuous function. Prove that f(x) = 2x + g(x) is uniformly…

A: The complete solutions are given below

Q: Prove f(x) = x2+4 is not uniformly continuous on R.

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Q: (11) (C) For the function f(x) = x° – x, determine whether the hypothesis of Rolle's theorem hold on…

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Q: Use the e- 8 property in Definition 19.1 to show that f(x) 1 is uniformly continuous on the interval…

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Q: Prove that f(x) = |x - 6| is continuous at all points c in ℝ.

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Q: 6. Let f be a continuous function on [0, 2] whose second derivative exists on (0, 2). Assume f(0) =…

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Q: 2. Find functions f,g: (0, 1) -+R with f continuous and g uniformly continuous such that neither fog…

A: Assume any two functions which are uniformly continuous. Now determine the values of f∘g and g∘f.…

Q: 4. If continuous function f:[0,1] U (2,3) → R is one-to-one, is f also continuous? 1

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Q: 1. Construct a continuous function f: I R on a bounded interval I with the following property: f is…

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Q: Find all values of a such that f is continuous on R: |x + 1 if x a

A: For a continuous function LHL ( left hand limt) should be equal to RHL (right hand limit)

Q: Let f be a strictly positive, continuous function such that |f(x)| < x for all real x. Does the…

A: if we have 0 ≤ g(x) ≤ h(x) on the interval [a, ∞) then if  ∫a∞h(x) dx  converges  ⇒  ∫a∞g(x) dx…

Q: 1. Use the definition to prove that the function in f(r) = 2r – 3 is uniformly continuous. 2. Use…

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Q: 3. Apply the three conditions of continuity to determine if the function f(x) = 2x²+5x+2 x²-x-12 is…

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Q: Show (in terms of  −δ) that a function f : [2,7] −→R be defined by f(x) = √x2+1 is uniformly…

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Q: 1. Determine whether k(x) is continuous at r = 3, where %3D x + 3, x 2.

A: Given  k(x)=x+3,  x&lt;0x2+1, 0≤x≤22x,   x&gt;2 To check the continuity we can use the graph as…

Q: 13. Let g be defined by V7x +2 V6x + 4 | g(x) = if x 2 - and x 2 x – 2 k , if x = 2 For what value…

A: The given functiong(x)=7x+2-6x+4x-2,if x≥-27 and x≠2k,if x=2

Q: 3 - x, x<2 Q.No.15: Is f(x) = {+2 x22 continuous at every x € R. x+1 xs1 -.

A: Continuity of a function.

Q: 2.23 Let 1 - x2 f(x): if x € Q, if r ¢ Q. Find all points p at which f is continuous. Prove your…

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Q: 2) Determines whether the following functions are uniformly continuous on the specified Intervals.…

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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) Discuss the continuity of piecewise function, x-7x+12 x 3 at x= 3 using the continuity test. (b)…

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Q: 7. Find the intervals on which the function is continuous, J (X) = +3)2+6

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Q: 1-x², x 2 at x = 2.

A: Given, fx=1-12x2, x&lt;23,    x=2x-2-1,   x&gt;2

Q: øz (@) is a continuous function of o in the range -o < @< ∞.

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Q: 19 If f(x) be continuous function for all real values of x and satisfies; x2 + {f(x) – 2 }x + 2/3 –…

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Q: Show that f(x) = 1/x2 is uniformly continuous on the set [1,∞) but not on the set (0, 1].

A: To show that f(x)=1x2 is uniformly continuous on the set [1,∞) but not on (0,1].

Q: Prove that there is no continuous function f : S² → S' such that f(-x) =-f(x) for all x € S².

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Q: 3. Let f(r) = 8 – 4x4/5. Carefully explain why f is continuous on [-1, 1), f(-1) = f(1), but there…

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