Let a < b be given. Give examples of continuous functions g, h : (a, b) → R such that (a) g is not bounded. (b) h is bounded but does not attain its bounds.

Q: Assume that a continuous function f:R → R is X- and µ-periodic, where A and u are two nonzero…

A: We know that between two rational number lies infinite many irrational number. But function is…

Q: Suppose that the function f:(a, b) → R is uniformly continuous. Prove that f: (a, b) → Ris bounded.

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Q: 10 f(x) = g(x): for a x+1 nction g?

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Q: Suppose that g: [a, b] R is a continuous function. Show that there exists a point c € (a, b) so that…

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Q: Show that a uniformly continuous function on (a, b) is bounded.

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Q: Every bounded continuous function attains its maximum and minimum values.

A: Here’s a sketch of one possible argument. Let u=supx∈[a,b]f(x). (Note that I allow the possibility…

Q: Let I be an interval and let f : I → R be a strictly increasing function such that f(I) is an…

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Q: Let X be Hausdorff and f : X → X be a continuous function. Show that {x : f (x) = x} is closed in X.

A: Let, .We show  is open and so F is closed.

Q: Let f: S¹→ S¹ be a continuous function. Proof or disprove the following statements : a) f has a…

A: Given that the  function f : S1 → S1 be a continuous function  where S1 = x,y∈ℝ2 : x2 + y2  =1 We…

Q: Which Theorem states: If f is continuous function on the open interval (a, b) then f attains an…

A: Since the extreme value theorem states that- If f is a continuous function on [a,b] , then f has…

Q: 5. Let f: Z → Z be defined by f(n) = n² + 1. (a). Is f a one-to-one function? Explain. (b). Is f an…

A: Given- The function is f(n)=n2+1 and f:ℤ→ℤ. Definition- Let a function from X to Y, One-to-one: If…

Q: Consider the function x - 3 for x4 By using the characterization of continuous function in terms of…

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Q: Let f:X → Y be a linear function, where X and Y are normed spaces over the field of real numbers.…

A: The given linear function: Where X and Y are normed spaces over the field of Real numbers. To prove…

Q: アー)。 The xy-plane above shows the graph of a continuous function f that is defined on interval [-2,…

A: We use the graph of f(x) and the definition of the function g(x) to answer the given question.

Q: Suppose f: R → R and g: R → R are continuous functions. True or false: The composite function fog is…

A: Given: f: ℝ→ℝ and g: ℝ→ℝ are continuous functions."The composite function f ∘ g is uniformly…

Q: Suppose f: R → R and g: R → R are continuous functions. True or false: If the composite function fog…

A: f:ℝ→ℝ and g:ℝ→ℝ is continuous.

Q: A function that is uniformly continuous on a set S is not necessarily continuous on S.

A: Given statement A function that is uniformly continews on a set S is not necessarily continews on S…

Q: Theorem 11. Let f be a continuous function on R such that f(x+y)-f(x)+fv) vx, yER. Show that f(x)3cx…

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Q: Suppose f: IR→ R and g: R → R and h: R→ R are continuous functions such that h(x) = f(x)g(x) for…

A: Using given information we have to tell that whether the  given statement is true or false.

Q: Show that for any function f : X → Y , there are two functions g : S → T and h : X → S such that (a)…

A: To Show: Show that for any function f : X → Y, there are two functions g: S → T and h: X → S such…

Q: Suppose f and g are continuous on a closed interval [a, b] such that f(a) > g(a) and f(b) < g(b).…

A: I have used intermidiate value theoram of continuous function.

Q: Metrics. Show that, for fixed x0 in E, the function  from E into ℝ+ is uniformly continuous.

A: Given:Let  be a metric space, and let ​ be a fixed element in .Objective:We want to show that the…

Q: Every function defined on a closed interval [a, b] must have a global maximum and a global minimum…

A: Find your answer below

Q: If f:X - Y is a continuous function then f- (V) is open in X for every open set V in Y. True False

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Q: • Assume f : A R continuous, and A open. Is f(A) open? • Assume f: A R continuous, and A compact. Is…

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Q: prove that any Continuouse function [a,b] en is measurable.

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Q: Let f and g be continuous functions on [a, b], and suppose that f(a) g(b). Prove that f(c) = alc)…

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Q: (b) Let X = (a, b]. Construct a continuous function f : X → R (set of real numbers) which is…

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Q: fa function f is continuous on the closed interval a, b, a, b E R, then f has an absolute maximum…

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Q: ) Let f be a continuous function that maps a finite closed interval [a, b] into itself. Prove that f…

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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: Exercise 3.3.10: Suppose f: [0,1] → [0,1] is continuous. Show that f has a fixed point, in other…

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