9. Find an example of an ƒ : (0, 1) → R which is continous but not uniformly continuous. Where exactly did we use the fact that [a, b] was a closed and bounded interval in the proof of the theorem?

Q: Mathematicians B. Constante and M. Sosa are having an argument. Professor Sosa claims that it is…

A: Professor Constante is correct.

Q: Suppose g [a, b] → R is a continuous function that takes only irrational values. Prove : that 9 is…

A: Use the Intermediate Value Theorem (IVT) and method of contradicts.

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Q: 2. Consider the function / with domain (-2, o0): 3|2 – 1| 2 + 27 - 4 if – 2 <z<1 f(z) = if r = 1…

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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: 46. Prove that there is no continuous function f: R→ R such that, for each c E R, the equation f(x)…

A: Suppose such continuous function  f  exists.  For each c in R, the equation f(x)=c has exactly two…

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: [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&amp;nbsp;p, for which f is…

Q: 2. Show that if f : R → R is continuous on R and f(r) = 0 for all r E Q then f(r) = 0 for all x E R.

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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: "Prove that the function f : R→ R with ƒ(x) = = is uniformly continuous."

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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: Let ƒ : R\{1} →→ R\{1} be defined by f(x) = x/(x - 1). Show that ƒ is bijective and find f-¹. Also,…

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Q: Suppose f : [a, b] → R is is continuous and such that fg = 0 for all continuous functions g : [a, b]…

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Q: (a) Assume f: S → R is uniformly continuous and S is bounded. Prove that the range of f is bounded.…

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Q: 6. Let S = {-1,0, 2, 4, 7}. Find f(S) (image of S under f) where f is from R to R for (a) f(a) = 1…

A: The

Q: 4. Prove that the function f: [0, ∞)→ R; f(x) continuous. = √ is uniformly

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Q: #2 Given A C R" and f : A → R", define T; = {(x, f(x)) E R" × R" | x E A}. The following questions…

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Q: If f : [0, 3] → R is continuous not constant, then its range be a closed, bounded interval?

A: Yes true.

Q: f(c) = g(c).

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Q: Determine if the requirements for Rolle's theorem are met by the function f(x) = 5x + +3 guaranteed…

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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: 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) Show that C(0, 1] C L'(0,1]. Compare |flsup and fl. Justify your answer. Generalize this to the…

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Q: Let n > 2 be a natural number. Prove that there is no continuous function q : C → C such that…

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Q: 1. If f:(0, 3) → R is continuous, must its range be an open (possibly infinite) interval?

A: Given statement is : If f:(0, 3) → R is continuous, must its range be an open (possibly infinite)…

Q: 1. Define f(x)= Prove that f: R→ R is continuous. if x ≥ 0 -r² if r ≤0

A: Given: fx=x2     if x≥0-x2 if x≤0 To prove: f is continuous.

Q: Let f: [1,1]→ R be a continuous function. Prove the following statements: (a) If there is c = [-1,…

A: Given that  be a continuous function.(a) We need to prove that, if there exist  such that , then…