What does it mean to say f : (a,b) →R is a bounded function? If a < b in R and f : (a,b) → R is uniformly continuous, prove it is a bounded function. Give examples of two functions f,g : R → R which are both uniformly continuous and where f is not bounded but g is bounded.

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

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Q: Show that if f: A-> B is continuous and A is compavt then f(A) is also compact

A: Let , f : A →B be continuous function. Here , A is compact set. We need to prove that , f(A) is also…

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: 4. Construct following three types of functions. (1) f: I R is continuous and I is bounded. But f is…

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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 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: 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&lt;x*&lt;b then…

Q: 6. Let f be continuous on [a, b]. Show ffdF = f(x) [F(b) – F(a)] for some x € [a, b].

A: This is a consequence of the intermediate value theorem

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: Let f : {1, 2, 3} → {a, b, c} be one-one and onto function given by f(1) = a,   f(2) = b and f(3) =…

A: Consider

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: 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: IR→IR and g: functions. True or false: If the composite function f o g is monotonic, then…

A: Given: f:R→R and g:R→R are the continuous function. To Find: To determine whether the following…

Q: 2. Suppose f : X →Y and g : Y → Z are functions, and g of :X → Z is injective. (a) Show that f is…

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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: 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: 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: This is real analysis question. Recall that if a function f : R → R is differentiable, then it is…

A: Let f: ℝ→ℝ be a differentiable function. (a) Suppose f'x≥1 for all x∈ℝ. Let K⊂ range (f) be a…

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

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

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Q: Define f: R -R by Sir +1 if r 4. Find all real values at which f(x) is continuous. Prove that your…

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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:C-R be a function where Cis a closed and bounded interval. Prove that if f is continuous then…

A: The given problem is to prove that the function is maps from closed and bounded interval to Real…

Q: Let a function f be analytic everywhere in a domain D. Suppose that f(z) is pure imaginary for all z…

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Q: Suppose that the function f: [0, 1] in R is continuous and thatf(x) &gt; 2      if 0&lt;=X &lt; 1.Is…

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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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Q: Let X, Y and Z be sets. A, B are subsets of Y. f:X → Y is a function. Prove the given statement if…

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Q: If f is a function from X to Y and A, B are subsets of X then .A f(A – B) = f(A) – f(B) .B f(AUB) =…

A: Given: f:X→Y and a,b⊆X, Find out: To choose the correct option.