A function f is of bounded variation on the closed interval [a, b] if there is a number Ksuch thatEls(2;) – f(xj–1)| < Kj=1for any partition P = {a = xo < x1 < • · · < xm = b}. Assume f is of bounded variationon [a, b]. Prove the following statements:%3D(i) The function ƒ is bounded on [a, b].(ii) The function f is integrable on [a,b].

Given that, A function f be a bounded variation on the interval a,b Then there exists a Kreal number…

Answered: A function f is of bounded variation on…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Let f be a function defined on [0,2], and f (x) = -x² + 1, Find L (p. f) only where p, is sequence of partition divide the interval into n equal parts.
2. Is the proof correct? Either state that it is, or circle the first error and explain what is incorrect about it. If the proof is not correct, can it be fixed to prove the claim true? Claim: If f : (0,1] → R and g : [1,2) → R are uniformly continuous on their domains, and f(1) = g(1), then the function h : (0, 2) → R, defined by h(æ) = { F(x) for x € (0, 1] | 9(x) for x e [1, 2) is uniformly continuous on (0, 2). Proof: Let e > 0. Since f is uniformly continuous on (0, 1], there exists d1 > 0 such that if x, y E (0, 1] and |æ – y| 0 such that if æ, y E [1, 2) and |r – y| < d2, then |9(x) – 9(y)| < €/2. Let 8 = min{d1, &2}. Now suppose r, Y E (0, 2) with x < y and |x – y| < 8. If x, y E (0, 1], then |x – y| < 8 < 81 and so |h(x) – h(y)| = |f (x) – f(y)|< e/2 < e. If x, y € (1,2), then |æ – y| < 8 < d2 and so |h(x) – h(y)| = |9(x) – g(y)| < e/2 < e. If x € (0,1) and y E (1,2), then |x – 1| < |x – y| < 8 < dị and |1 – y| < |æ – y| < 8 < 82 so that |h(x) – h(y)| = |f (x) – f(1) + g(1) –…
1. Define a function f : Z x Z → Z by f(m, n) = 2n – 4m. (a) Prove that ƒ is not one-to-one. (b) Prove that the range of f is equal to the even numbers. In other words, show rng f = {2k|k E Z}, by using the usual methods for proving an equality of sets.
Use the formal definition to show that ww w the function f(x) = x3 + 4x² is not uniformly continous on the interval (-∞, 0)
There is a number c in the interval [a, b] such that f(c) is the absolute minimum of f on [a, b]
Let f be a function defined on [0,2], and f (x) = -x² + 1, Find L(Pn. f) only where pn is sequence of partition divide the interval into n equal parts.
Let f(z) be an entire function. If |ƒ'(z)| ≤ |z| for all z € C, prove that there exists a, ß E C such that f(z) = a + ßz² for all z E C and |ß| ≤ 1.
Let ℓ be a line and let f : ℓ → R be a function such that PQ = |f(P) - f(Q)| for every P, Q ∈ ℓ. Prove that f is a coordinate function for ℓ.

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