Let f, (x) uniformly converge to f(x) on [a, b). If f„(x) € R[a, b), then f(x) E R[a, b] O True O False

Q: ..... Q2: (a) Let a ,b ER, and a < b. Prove that 3s ER-Q,a <s<b. (b) Let o ACR be bounded below.…

A: I assume that you have already proved that between any two reals there exist an rational number.

Q: The function f(x) = x-2 is uniformly continuous on ]2,4]. 12, 4]. True False

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Q: 5. Let f.(x) = a". Show that "0 uniformly on (0, . handwriting

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Q: 3. Let f : [2,5] → R such that f(x) et Is f uniformly continuous? Justify your answer. Answer:

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Q: 3. Let f be continuous on [-1, 4] and differentiable on (-1,4). If f(-1) = –1 and f(4) = 4, then the…

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Q: Suppose ƒ : (a, b) → R is differentiable and that ƒ' : (a,b) → R is bounded. Prove that f: (a, b) →…

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Q: Consider the regular subdivsIon of the interval [a, b] as a = x0 < x1 < x2 < x3 < x4 = b, with the…

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Q: Prove that if f: A to R is continuous at x=c, then for all (x_n) converging to c, it follows that…

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Q: 1. a. Suppose F is of bounded variation on [a,b], then |F'| = VF' a.e b. F is absolute continuous…

A: It is given that Suppose F is of bounded variation on [a,b]

Q: Show that the function f(x): = √/x² is uniformly continuous ON A:= [1,00), but that it is not…

A: To show that f(x) is uniformly continuous on A, we use definition of uniform continuity and to show…

Q: converges weakly to x in a normed linear sp

A: Given that, xn converges weekly to x in a normed linear space X We have to show that Every…

Q: Exercise 4: Suppose f is differentiable on (2,6) and continuous on [2,6. Suppose further that 1 < f'…

A: Given:f is differentiable on (2, 6) and continuous on [2, 6].

Q: Show that f„(x) = x" (1– x") converges pointwise V x € (-1,1]. Is f„(x) converges uniformly?

A: When a sequence includes a variable, it becomes a sequence of function. A sequence of function can…

Q: 19) Prove or find a counter example to the following statements: a) f is continuous on (a,b) implies…

A: Step 1: Explanation for question

Q: A monotone sequence is convergent.

A: 1) If f is continuous on A then fn is also continuous  on A So The statement is true

Q: 2.) Determine the set of points at which the function is continuous. а. F(x,у) — 1+x²+y² 1-x2-у? ху…

A: Here, to find the set of points at which the function is continous.

Q: Suppose that f is continuous on that 9< f(4) < 21. [0, 4], f(0) = 1, and 2 < f'(x) < 5 for all a in…

A: Here in this question we have a function which is both continuous and differentiable in the given…

Q: 3. Consider the usual exponential function f(x) = e. (a) Without using a calculator, use linear…

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Q: 5) Suppose that f is continuous on (-00, 00). How many non-differentiable points could f have? Is it…

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Q: Suppose that X P(X < 3Y). unif(-2, 2) and Y ~ unif(-1,2) are independent continuous RVs. Find P (|X…

A: Given: X ~ U (-2, 2) Y ~ U (-1, 2) X and Y are independent.

Q: 4. Prove fx=x2+4 is not uniformly continuous on R

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Q: post office sells 4-cent sheep stamps and 2-cent sunset stamps. Cole bought 5 stamps spent exactly…

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Q: Suppose that fand gare continuous functions such that f(z)dz = 2and s(z)dz = 3, find (2f(2) +…

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Q: 22) which of the following functions is uniformly continuous on the giuen set? a) flx)= x 3 on 10,1)…

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Q: 55. Let F be a function given by Sx2 + 1, for x s 2, 2x + 1, for x >2. F(x) = a) Verify that F is…

A: We have Fx=x2+1 ,  for x≤22x+1 , for x&gt;2 We have to verify   (a) Function is continuous at x=2…

Q: be a sequence of F-integrable functions on [a, t ow f is F-integrable and

A: Let  be a sequence of integrable functions whose choose  are arbitrary.Since  is uniformly…

Q: Suppose f ond 9 are continuous functions on [0. b] ond thand gee)>0 for all x E [0,b]. Prove that…

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Q: For i = 1, 2, ..., let X; ~ U (0, 4), and Y; ~ N (2, 4), independently. Find, E (X;) E (Y;) the…

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Q: the 18) use definition to deformine whether or not given functions are uniformly continuous a) for X…

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Q: 11. Prove that if f is continuous on (a, b] and lim,a* f(x) exists, then fis uniformly continuous on…

A: The theorem is done .

Q: 22. Let fn R→→ R be a sequence of functions such that f'(x), f"(x) are defined for all x € R.…

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Q: Suppose that 0.3210 approximates 0.3245. How many significant digits does the approximate have? O 3…

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Q: If a function f(x) is continuous on (a, b) and differentiable on (a, b), then the Mean Value Theorem…

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Q: Let T: (X,d) (Y,d) be a continuous function for all n. If {T,} converges uniformly to T on X, then T…

A: Given Tn:X,d→Y,d^ is a continuous function for all n and Tn converges uniformly to T on X. We have…

Q: O f:xYis a closed function if fCA) <PCÃ)VAEX

A: Given that The aim is to show that,

Q: 3. Define f(x) so that S(2) = {ap(). f(x) = 0, |피 1 (a) Explain why the function f(x) must be…

A: As per our guidelines we can answer only 3 subparts. please ask others seperately.

Sequence and Series

A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence.  It was invented by German mathematician Carl Friedrich Gauss.