A function f : R R is said to be additive if f(x + y) =

Question:

A function f : R → R is said to be additive if f(x + y) = f(x) + f(y) for all x, y in R. Prove that if f is continuous at some point x0, then it is continuous at every point of R.

Fantastic news! We've Found the answer you've been seeking!

Step by Step Answer:

Related Book For book-img-for-question

Introduction to Real Analysis

ISBN: 978-0471433316

4th edition

Authors: Robert G. Bartle, Donald R. Sherbert

See More Books

Question Posted: Apr 07, 2016 10:59 AM

See More Questions

A function f : R R is said to be periodic on R if there exists a number p > 0 such that f(x + p) = f(x) for all x R. Prove that a continuous periodic function on R is bounded and uniformly...
If h(x) := 0 for x < 0 and h(x) := 1 for x > 0, prove there does not exist a function f : R R such that f(x) = h(x) for all x R. Give examples of two functions, not differing by a constant, whose...
Use a picture like the one that appears in the discussion of the Chain Rule to answer: if a function f : R R has an inverse, what's the relationship between how the function-locally,...
A regression analysis resulted in the following information regarding a dependent variable (y) and an independent variable (x). n = 10 x = 55 y = 55 x2 = 385 y2 = 385 xy = 220 Refer to Exhibit. The...
Hannah Byers and Kathleen Taylor are considering the possibility of teaching swimming to kids during the summer. A local swim club opens its pool at noon each day, so it is available to rent during...
A73 kg man stands on a level bridge of length L. He is at distance L/4 from one end. The bridge is uniform and weighs 2.7kN what are the magnitudes of the vertical forces on the bridge from its...
A square loop of wire has a perimeter of $4.00 \mathrm{~m}$ and is oriented such that two of its parallel sides form a $25.0^{\circ}$ angle with the horizontal. A uniform horizontal magnetic...
YellowCard Company manufactures accessories for iPods. It had the following selected transactions during 2012. 1. YellowCard provides a 2-year warranty on its docking stations, which it began selling...
9. An arrow 2.5 cm high is placed at a distance of 25 cm from a diverging mirror of focal length 20 cm., Find the nature, position and size of the image formed. 10. The image formed by a convex...
Suppose that Bigco is currently trading for $100 per share. We know that in one year Bigco stock will sell for either $150 per share (good day) or for $75 per share (bad day). No other prices are...
Determine the points of continuity of the following functions and state which theorems are used in each case. (a) f(x) := x2 + 2x + 1/x2 + 1 (x R), (b) g(x) := x + x (x > 0), (c) h(x) := 1 + |sin...
Suppose that f is a continuous additive function on R. If c := f(1), show that we have f(x) = cx for all x R. [First show that if r is a rational number, then f(r) = cr.]
When a dilute solution of 6-phenylhexanoyl chloride in carbon disulfide was slowly added (over a period of 8 days!) to a suspension of aluminum chloride in the same solvent, it yielded a product A...
Software tech inc. is planning to buy a new machine and replacing an old machine. Old machine was purchased 10 years ago and is fully depreciated but it can still be used for the next 5 years....
You hold a plastic bottle of water upright in your hand, and carefully poke a small hole in the bottom of the bottle with a nail. Assume viscosity can be neglected. Use g = 9.80 m/s 2 . Part A: The...
You are needing to find out how much the installment payment might be on a car valued at $35,000. The interest rate on the 10 year installment is 4.75% per year. You will be required to make your...
Hawk Corp. had the following transactions in the current year: Short term capital gain $10,000 Short term capital loss -$2,000 Long term capital gain $4,000 Long term capital loss -$15,000 If Hawk...
Submit a paper detailing types of audit reports for both issuers and nonissuers, highlighting the differences between the requirements for the audit reports for issuers and nonissuers. Be sure to...
Deloitte interviewed a number of prominent organisations about their vision of XBRL. What is XBRL and what value does XBRL bring to an organisation? Access quite a good video clip...
On 1 July 2018, Parent Ltd acquired all the shares of Son Ltd, on a cum-div. basis, for $2,057,000. At this date, the equity of Son Ltd consisted of: $ 1,000,000 Share capital 500 000 shares...
Represent the function (a) By its Maclaurin series, and state where the representation is valid; (b) By its Laurent series in the domain 1 fa)
Write the two Laurent series in powers of z that represent the function In certain domains, and specify those domains. 2tDEJ
(a) Let a denote a real number, where 1 (b) After writing z = eiÎ¸ in the equation obtained in part (a), equate real parts and then imaginary parts on each side of the result to derive...
Can you elucidate the principles underlying genome organization and chromatin structure, including the roles of topologically associating domains (TADs) and chromosomal territories in gene regulation...
State three ways of using artificial intelligence and machine learning in marketing and explain each use case with two examples.
discuss the principles of population genetics, including allele frequency distributions, Hardy-Weinberg equilibrium, genetic drift, and natural selection, and how these concepts inform our...

A function f : R R is said to be additive if f(x + y) =

Question:

Step by Step Answer:

First show that f0 0 and fx fx for all x R then ...View the full answer

Introduction to Real Analysis

Students also viewed these Calculus questions