Question: Suppose that f : R (0, ) satisfies f(x + y) = f(x)/(y). Modifying the outline in Exercise 3.3.8, show that if f is

Suppose that f : R → (0, ∞) satisfies f(x + y) = f(x)/(y). Modifying the outline in Exercise 3.3.8, show that if f is continuous at 0, then there is an a ∊ (0, ∞) such that f(x) = ax for all x ∊ R. (You may assume that the function ax is continuous on R.)

Step by Step Solution

★★★★★

3.46 Rating (178 Votes )

There are 3 Steps involved in it

1 Expert Approved Answer

Step: 1 Unlock

a f0 f0 0 f0f0 f 2 0 implies f0 0 or 1 But f has range 01 s... View full answer

Question Has Been Solved by an Expert!

Get step-by-step solutions from verified subject matter experts

Step: 2 Unlock

Step: 3 Unlock

Document Format (1 attachment)

741-M-N-A-D-I (240).docx

120 KBs Word File

Students Have Also Explored These Related Numerical Analysis Questions!

Suppose that f : R R satisfies f(x + y) = f(x) + f(y) for each x, y R. a) Show that f(nx) = nf(x) for all x R and n Z. b) Prove that f(qx) = qf(x) for all x e R and q Q. c) Prove that f is...

Suppose that f : R R. If f" exists and is bounded on R, and there is an 0 > 0 such that |f'(x)| > 0 for all x R, prove that there exists a > 0 such that if |f(x0)| 0 for some x0 R, then f...

Suppose that f : [a, b] R is continuous and that for all integers n > 0. a) Evaluate ba P(x) f(x) dx for any polynomial P on R. b) Prove that ba|f(x)|2 dx = 0. c) Show that f(x) = 0 for all x [a,...

Suppose that f : R ---+ (0,00) satisfies f(x + y) = f(x)f(y). Modifying the outline in Exercise 8, show that if f is continuous at 0, then there is an a E (0,00) such that f(x) = aX for all x E R....

8. Suppose that f : R ---+ R satisfies f(x + y) = f(x) + f(y) for each x, y E R. (a) Show that f(nx) = nf(x) for all x E Rand n E Z. (b) Prove that f(qx) = qf(x) for all x E Rand q E Q. (c) Prove...

can you show me How to do both of these sorry the pictures weren't coming out before. 1.suppose that f:R ?>R satisfies f(0)=0 and f'(x)=1 for every x, show that for every x, f(x)=x 2) give an example...

MATH0019 Homework 2 Due: October 28, 2022 Basic 1. Consider polar coordinates in the ry-plane given by the equations r = vx2 +y?, 0 = tan-'(y/x), which can be solved for r, y to give I = roos0, y =...

University of California, Los Angeles Department of Statistics Statistics 100C Instructor: Nicolas Christou Homework 3 EXERCISE 1 Answer the following questions: a. Suppose y1 , y2 , . . . , yn are...

2. (a) Let X be a continuous random variable taking values in (a, b) where a ( RU {-co} and be RU foo}. (This simply means that we allow the sample space to be (a, co), (-0o, b), or .) Suppose that...

Design for Experiments: The strain gage shown at right is designed for measuring shear stress and torque in a rotating shaft. Since the gage needs to be mounted to the rotating shaft whereas the...

(a) What is the entry-level salary for the job you are seeking? (b) What salary range do you plan to seek in your job search? (c) What is this range based on?

How do the empirical rule and the Chebyshev rule differ?

(Market for Pollution Rights) The following graph shows the market for pollution rights. a. If there are no restrictions on pollution, what amount will be discharged? b. What will be the quantity...

4. For each of the following, find IIs w, where n is the outward-pointing normal. (a) S is the topological boundary of the three-dimensional region enclosed by y = x2, Z = 0, z = 1, y = 4, and w =...

3. For each of the following, evaluate IIs F nda, where n is the outward-pointing normal. (a) S is the topological boundary of the rectangle [0,1] x [0,2] x [0,3] and F(x, y, z) = (x + eZ , y + eZ ,...

2. For each of the following, evaluate Ie w. (a) C is the topological boundary of the rectangle [a, b] x [c, d], oriented in the counterclockwise direction, and w = (f(x) + y) dx + xy dy, where f :...

2. Exchange options. Suppose that: The stock of Exxon Mobile Corp. (XOM) is currently trading at 76.38. It is expected to pay an 80 cent dividend at the end of each of the next four quarters. It has...

What is a key feature of the Internet of Things ( IoT ) ? 1 a . It limits connectivity to wired networks only b . It involves connecting everyday devices to the Internet c . It excludes home security...

5. Howton & Howton Worldwide (HHW) is planning its operations for the coming year, and the CEO wants you to forecast the firm's additional funds needed (AFN). The firm is operating at full capacity....