2. Let f(x) = 1 for rational x and f(x) = 0 for irrational x. Show that if F is increasing and continuous on [a, b] and F(a) < F(b), then f is not F-integrable on [a, b].

Q: interval [a,b] where it is NOT integrable. f(x)=           ,   x∈

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Q: The function f is continuous on [0, 4]. f(0)= -2 and f(4) = 5. Must there exist a number C between 0…

A: Given,         The function f is continuous on 0,4. f0=-2 and f4= 5.         There exist a number c…

Q: the function f is not differentiable at x=0.

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Q: Prove that if f is a continuous function on a closed interval [a, b], then

A: We have to prove the given inequality for a continuous function on a closed interval [a,b].

Q: Let Ax) = 2x - 7x +4x. %3D a. On what interval(s) is f'(x) > 0? b. On what interval(s) is fx)…

A: Given: fx=2x3-7x2+4x   The Power law: ddxxn=nxn-1

Q: 3. Let A be the set of all functions ƒ : 0, 1] → R that are continuous in the interval [0, 1].…

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Q: If f is differentiable on R and f(0) = 1,|f0(x)| < 1,∀x ∈ R. Show that |f(x)|≤|x|+ 1,∀x ∈ R.

A: Note that |f0(x)| &lt; 1 should have been |f'(x)| ≤ 1. Otherwise, the given inequality won't be…

Q: Let f(x) = (Vx + b² - b)/x, b > 0. What is the domain of f? (Select all that apply.) O (-∞, ∞)…

A: f(x)=x+b2-bx The given query is to find the domain of f(x) Also, to define the function such that it…

Q: Left- and right-continuity a. Is hlx2 = Vx - 9 left-continuous at x = 3? Explain. b. Is h1x2 = Vx -…

A: To Determine: find left hand and right hand continuity at x=3 and explain this Given: we have a…

Q: Suppose that f is a function which is integrable on the entire real line, and that a and b are real…

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Q: n a and b with f(c) = N. I thi

A: Mean value theorem: Rolles theorem If a real-valued function f is continuous on a proper  [a, b],…

Q: Let f: IR be a function. (a) Prove that if f is differentiable at some c I for which f(c) ‡ 0, then…

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Q: Use the e-6 definition of continuity to prove that the following functions are continuous at the…

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Q: Show that if f"(x) < 0 for x ∈ [a, b], then 1/2[f(a) + f(b)] < favg < f[(a+b)/2]

A: Consider the given information. f"(x) &lt; 0 for x ∈ [a, b], then 1/2[f(a) + f(b)] &lt; favg &lt;…

Q: Let f (x) = x2 if x is rational and f (x) = 0 if x is irrational. a) Prove that f is continuous at…

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Q: A function is said to be Hoelder a continuous on [a, b] iff |f(x) – f(x')| 0. Show that if a > 1…

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Q: For any R > 0, show f(a) is continuous at R when f(a) sqrt (x) is continuous on (0, +infinity) at…

A: Let R&gt;0.  Suppose fx=x is continuous on 0,∞ at R=b To prove that fx is continuous on R Let b&gt;0…