Prove that f (x)= x-1/2 is :(i) not sectionally continuous in every finite interval in the range x> 20.(ii) of exponential order as x→ o. Is L {x-12; p} exists ? If it exists find the value

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Answered: Prove that f (x) = x-12 is : (i) not…

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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1. Let a be a real number and f be defined by { What are the values of a for which f is continuous at x = 0? f(x):= eax x ≥ 0 ax+1, x < 0.
Determine whether the following statements are true or false. If true, provide a proof; if false, provide a counterexample. (a) Let f, g, h be continuous on the interval [a, b]. If f(a) g(b) > h(b), then there exists c = [a, b] such that f(c) = g(c) = h(c). (b) Suppose that f and g are continuous on R. If 0 ≤ f(x) < g(x) for all X, then there is some x ER such that f(x)/g(x) is the maximum value of f/g. (c) If f is continuous on R, then f is bounded.
Suppose f: IR→IR is a monotonic function. True or false: If f is differentiable, then f must be uniformly continuous. True False
Let f be a real-valued continuous and differentiable function. Let function g be defined by g(x) = f(|x| + 2). A student presents the following proof to show that there exists a real number c € (-1, 1) such that g/(c) = 0. (1) Since f is a continuous function, so is g over the interval [-1, 1]. (II) Since f is differentiable, so is g over the interval (-1, 1). (III) It is evident from the definition of g that g(−1) = g(1). (IV) If the above conditions hold, then by Rolle's theorem, there exists dg(x) c = (-1, 1) such that gl (c) dx |x=c Which statement about this proof is correct? = 0. Step (1) does not hold, and hence Rolle's theorem does not apply. Step (II) does not hold, and hence Rolle's theorem does not apply. Step (III) does not hold, and hence Rolle's theorem does not apply. Step (IV) does not hold, and hence the conclusion is false. The proof is completely correct, and the conclusion holds.
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) –…
4. `s.] Find the number a and b at which f is continuous everywhere? x² -1 if x 2 Nt
Use the three-steps definition of continuity to show that the function f(z) = z² - 2z+ (1 + i) is continuous at zo = 3 - i.
(5) Let f: RR be twice-differentiable f(x) ≤0 and Prove that f is constant on R. on R. Suppose that f"(x) ≥0 for all x = R
Use FTC I to prove that if |f '(x)| ≤ K for x ∈ [a, b], then |f (x) − f (a)| ≤ K|x − a| for x ∈ [a, b].
h5. Prove that f ( x ) = x ⋅ | x | is continuous at all points c in ℝ.
Let f be defined by f(q)=3q for q rational and f(t)=t2 for t irrational.. b) is f continuous at 3? prove your answer, remember to prove true you MUST prove it to be true, not use a specific example.

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Prove that f (x) = x-12 is : (i) not sectionally continuous in every finite interval in the