6. Suppose that f:R → R is uniformly continuous so that for some o > 0, and for all r, y ERI/(1) – f(u)| <1 whenever |r – y| < 8.If f(0) = 0, Prove that for every a > 0f(2)| <1+ r/8.Hint: For each r> 0 show that |f(r)| <1+ng, where n, is the greatest integer not exceeding a/6.7. Define the sequence (a,) as follows: For every natural number n, leta, = [1.7+ (-1)"]".Does lim sup a, exist? What about lim inf a,? Find the set of subsequential limits for the sequence (a,).

The objective here is to prove: if f(0)=0then for every x>0 fx<1+xδ Given: f:R→R is uniformly…

Answered: 6. Suppose that f: R -R is uniformly…

6. Suppose that f: R -R is uniformly continuous so that for some 6 > 0, and for all 1, y € R I/(=) – f(y)| <1 whenever |r – y| < 6. If f(0) = 0, Prove that for every z>0

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Similar questions

1. Define f(x) = Prove that f: R→ R is continuous. x² -2² if x ≥ 0 if x ≤0
If |fn(x) – fn(y)l 0 and all x, y in a compact interval, show that {fn} is uniformly equicontinuous.
4. Let f: [-1, 1] → R be a continuous function. Prove the following statements: (a) If there is c € [-1,1] such that f(c)f(−c) < 0, then there is de R such that f(d) = 0. (b) If ƒ([-1, 1]) = (-1, 1), then f is not continuous.
1. Suppose f is continuous on R and f satisfies f(x) + f(2x) = 0 for all x E R. Prove that f = 0 on R.
2. Let f : R → R be a continuous function. Suppose that (f o f) (x) = x for every x E R. (a) Prove that there exists a point c ER such that f (c) Hint: Consider the function g (x) = f (x) – x for every x € R. = C. (b) Give an example for a function satisfying the above requirements, and such that f id, -id , where id is the identity function.
3. A function f: D→ R is called a Lipschitz function if there is some non-negative number C such that f(u)-f(v)| ≤ cu-v for all points u and v in D. Prove that if f: R→ R is a Lipschitz function, then it is uniformly continuous. € 4. Define f(x)=√ for all x ≥ 0. Verify the e- & criterion for continuity at z = 4 and at z = 100. (Hint: First show that for x ≥ 0, zo > 0, √ √ ≤X
15. Let f. g : R → R be continuous at a point c, and let h(x) := sup (f (x). 8(x)} for x € R. Show that h(x) = (f(x) + g(x)} + }If(x) – g(x)| for all x € R. Use this to show that h is continuous at c.
If f is a continuous function on R with f(1) > 0 and f(4) < 0, then there exists a number c between 1 and 4 such that f(c) = 0. Select one: OTrue O False Fi
3. Assume that f(x) is continuous on [a, b] and x₁,x2,...,n are arbitrary numbers taken from [a, b]. Show that there exists c E (a, b) so that f(c) 1 · [ƒ(x1) + ƒ(x2) + ... + f(xn)]. n == Hint. Use Intermediate and Extreme Value Theorems.
1. Use the e-6 definition to prove that f(x) = r² is uniformly'continuous on [a, b).
Suppose f is continuously differentiable on an interval (a, 6). Prove that on any closed subinterval [c, d] the function is uniformly dif- ferentiable in the sense that given any 1/n there exists 1/m (inde- pendent of ro) such that |f(x)- f(xo)-f"(x0)(x-x0)| < |x- xo|/n whenever r – xol < 1/m. (Hint: use the mean value theorem and the uniform continuity of f' on [c, d].)