5. Suppose that f: R→ R is continuous. and f(x) > 0.. Prove that there is an in- terval 7 = (20-20+1), where n is natural number, such that f(x) > 0 for all r in I.

Q: Let f: R R be defined as f(x) D 1 + x2 Let /= [- 1, 2), then f(1) = Select one: а. 1] b. (0, d. None…

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Q: 5. Let f : R → R be defined by f(x) = { 0 if r € Q, x2 otherwise. Prove that f is continuous at 0…

A: Here we prove the problem using the definition of continuity and a theorem.

Q: Suppose that f is a continuous function on [0,2] such that f(0) = f(2). Show that there is a real…

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Q: 3. Let f, g to be two continuous functions on [a, b] and f(a) ≥ g(a), f(b) ≤ g(b). Prove that Exo €…

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Q: b) Give an example to show that the product fg need not be uniformly continuous.

A: Given: f:A→B, g:B→C are uniformly continuous.To show: b) The product fg need not be uniformly…

Q: [1+2p(x-1)+(x-1)² x≤1 x>1 [qx+p -) -) Find the values of q, in terms of p, for which fis continuous…

A: Given:f be a function defined by:a. To find the value of q in terms of p, for which f is…

Q: 2. Suppose that f is an increasing function on the interval [a, b]. Let F(t) = %3D a.) Argue that if…

A: suppose that f is an increasing function on the interva a,b let Ft=∫0tfxdx (a) if a<x*<b then…

Q: Suppose that f: A → R is continuous at a number a E A with f(a) > 0. Show that there exist numbers L…

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Q: 7. Suppose that f: R → R is continuous at c and that f(c) 0 such that f(x) < 0 in the…

A: Recall the definition of continuity around c .And then choose an ∈&gt;0 to get a δ&gt;0…

Q: → R be a function defined by 3x² + x+5 f(x) = not continuous at x = = 0.

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Q: Is there a continuous function such that - f: [1,2] U [3,4] → R - f(1) * f(4) < 0 - f(x) not egual 0…

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Q: 6. Suppose that ƒ is differentiable on (a,b). interval [a, b]? Justify your response for find a…

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Q: Solve the D. E. to the I. V. P. sinx(e-y + 1)x' – cosx = 1, y(0) = 0 %3D %3D (1 – e') (1 + cos.x) =…

A: We will find out the required solution.

Q: 11. with f(-1)=6, f(1) = 8 and f(6) = 3. Intermediate Value Theorem? Let f be a function that is…

A: Given:f be a function that is differentiable on the closed interval -1, 6 such…

Q: Un which of the following intervals is 1 f(x) = continuous? Vx-2 (2, + ) (-a, +) [1,2)

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Q: Let nEN be n >2. Determine all entire functions f that satisfy f(z") = \f{z)l" for all z EC.

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Q: Let G=C-{ZER: Analytic function f:G-C such that z = (f(z))" for all z EG z≤0} and let n be a…

A: Analytic function

Q: 15) Prove that if ffx) is continuous at x=0 then so is V f(x) provided that fix) zo for all x in the…

A: Step 1: Step 2: Step 3: Step 4:

Q: 55. Let F be a function given by Sx2 + 1, for x s 2, 2x + 1, for x >2. F(x) = a) Verify that F is…

A: We have Fx=x2+1 ,  for x≤22x+1 , for x&gt;2 We have to verify   (a) Function is continuous at x=2…

Q: 3. For m₁ and m₂ numbers, with m₁ { Prove that the function f:R → R is continuous but not…

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Q: There is a partial insurance policy that pays $80,000 upon the bad drought state (and $0 in the…

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Q: 4. Give an example of a continuous function f : (0, 1) → R that cannot be uniformly approximated by…

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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: Let f(x)=1/x. Show that f is bounded from above on any interval (α,1), where α>0, but the function…

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Q: Describe all extensions of the identity map of ℚ to an isomorphism mapping ℚ(3rd root of 2) onto a…

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Q: Prove: f(x) = 1+x² is Lipschitz continuous in R.

A: Step 1: Step 2: Step 3: Step 4:

Q: Prove that f: R→ R is continuous. 2. Suppose that the function f : [0, 1] → R is continuous and that…

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Q: Let f, g to be two continuous functions on [a, b] and f(a) ≥ g(a), f(b) ≤ - Prove that co € [a, b],…

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Q: -2 x<-1 Q1)a) For what values of a and b is g(x) = |ax - b - 1<x< 1, 3 x21 Continuous at x=1

A: From  the given problem:   gx=-2x≤-1ax-b-1≤x&lt;13x≥1     As we know that a function f(x) is…

Q: 3. Define f(x) so that S(2) = {ap(). f(x) = 0, |피 1 (a) Explain why the function f(x) must be…

A: As per our guidelines we can answer only 3 subparts. please ask others seperately.

Q: Prove that the function f from Nx N to N defined by f(m, n) = 2m3 is one-to-one but not onto.

A: