(a) Suppose f: (a, b) → R is continuous. Show that if f(a+) and f(b) are finite(i.e. <∞), then f is bounded on (a, b). Argue whether the converse is true.

Q: If f is differentiable at a, then f is continuous at a.

A: If a function fx is differentiable at x=a then function must be continuous at x=a. But a function…

Q: x(x²+3) Q7. Consider the function f : [0, 1] → R, → √√4+24 Explain why (a) f continuous, (b) f is…

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

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Q: 2. Show that if f is continuous on [a, b] with f(x) > 0 for all a ≤ x ≤ b, then 1/f is bounded on…

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Q: Let f : R" → R be continuous. Prove or give a counter example: if O C R" is open, then f(0) is open.

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Q: Let F be a continuous function on [a, c} , where a < b< e, and F(z) dz = 6, F(2) dz = 9 A. Evaluate…

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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: = √x + 2,-2 ≤ x ≤2 ax²-bx, x+1 x-2 ) 2 3

A: Topic:- calculus

Q: Q2: If f(x) and g(x) are continuous on [a,b]and if f = [g Prove that 3c e[a,b] such that f(c) g(c).

A: From the given information. The functions f(x) and g(x) are continuous functions on [a,b]. The…

Q: Let f : R" → R be continuous. Prove or give a counter example: if O c R" is open, then f(O) is open.

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Q: Let f : R" → R be continuous. Prove or give a counter example: if O c R" is open, then f(0) is open.

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Q: 3. Let f, g : [a, b] → [c, d] be cotinuous functions. Suppose that f is surjective. Prove that there…

A: We will use of intermediate value theorem to proof this

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: 1. Show that if f [a, b] → R is increasing, then f is of bounded : variation.

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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: 1. Show that if f [a,b] → R is increasing, then f is of bounded variation.

A: Given f: a, b → ℝ is increasing function.

Q: Suppose f : R → R is a continuous function with the property that f(x) = 0 for every x E Q. Prove…

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Q: 2: If f(x) and g(x) are continuous on (a,b]and if f =[s . Prove that 3cela,b] such that f(c)=g(c).…

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

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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: 1. If f(x) is bounded on [a,b], then f f(x) may not exist. Prove this statement theoretically or…

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Q: 3. Let z € R" be denoted by x = (₁, 2,...,n) and || x ||:= √√x² + x² + +x². Let f: RR be defined by…

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Q: Suppose that f :C → C is integer and that ƒof = f. What can you say about f ? Justify your answer.…

A: Please find the answer in next step

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

A: Given function f:01→R f is not continuous, but f satisfies the conclusion of extreme value theorem

Q: Theorem 18.1 states that "Let f be a continuous real-valued function on a closed interval [a, b].…

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Q: f Let f:(R,1,) - (R,T,), f(x) = x². Find f-(a, b) and then show that is continuous on R.

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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: Suppose that f and g are both continuous functions. Also f, g € [a, b] and that f= fg. Prove that 3c…

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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 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: Suppose that ci < c2 and that f takes on local maxima at ci and c2. Prove that if f is continuous on…

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Q: 4. (a) Let I = [a, b] and let f and g be bounded functions on I to R. Suppose that f(x) < g(x) for…

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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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