Question: A function f : R R is said to be additive if f(x + y) = f(x) + f(y) for all x, y in

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.

Step by Step Solution

★★★★★

3.42 Rating (168 Votes )

There are 3 Steps involved in it

1 Expert Approved Answer

Step: 1 Unlock

First show that f0 0 and fx fx for all x R then ... 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)

829-C-F-M (372).docx

120 KBs Word File

Students Have Also Explored These Related Calculus 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, 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,...

I know this proof is correct but could someone please answer my questions in black ink for all x E R . A function f: R - R is said to be periodic if there exists a number k > 0 such that f(a + k) =...

Problem 5. Which of the fottowing are true and which are false? 1. {{@}} U (0 = {9, {0}} 2- {{@}} U {9} = {0: {9}} 3- {9, {0}} {{9}}, {{0}}} = {9} 4- {9, {0}} {{9}}, {{0}}} = {{9}} 5. For sets A,B,...

Could someone please check my work and let me know if it's correct I'm very unsure of my work Please state all definitions and theorems that you will need: Thearea 5.4/ Suppose of: D - R is...

24) A function f : R - R is said to be odd if f(-x) = -f(x) for all x E R. The graph of f (x) is: symmetric about the x-axis. b) symmetric about the y-axis. c) symmetric about the origin. d) None of...

Could someone please check my work Please state all definitions and theorems that you will need: Thearea 5.4He Suppose of: D ->Ro is continuous on a compact set is . Then of is uniformly continuous...

Could someone please help me with this problem A function f: R - R is said to be periodic if there exists a number k > 0 such that f(a + k) = f(a) for all a E R. Suppose that f : R - R is continuous...

The controller of Loran Industries has collected the following monthly expense data for use in analyzing the cost behavior of maintenance costs Instructions(a) Determine the fixed and variable cost...

Let L: P1 P1 be the linear operator defined by L(at + b) = -bt - a. Find, if possible, a basis for P1 with respect to which L is represented by a diagonal matrix?

TB MC Qu. 20-75 (Algo) In 2024, internal auditors discovered that... In 2024, internal auditors discovered that Fay, Incorporated, had debited an expense account for the $1,595,000 cost of a machine...

Which of the following is a way companies attempt to influence public policy and law? a. Press relations b. Events c. Lobbying d. Publicity

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...

Question 1 9 of 3 5 George invested his savings in shares of a new company. The company's share price dropped by $ 2 . 2 5 at the end of the first year, and by a further $ 2 . 7 5 at the end of the...

Regarding Harvard business school case 'The B.F. Goodrich-Rabobank interes trate swap' is someone can answer below question? 5. Goodrich a. Compare 30-year fixed rate in the US market and the fixed...

Judy Johnson is choosing between investing in two Treasury securities that mature in five years and have par values of $1000. One is a treasury note paying an annual coupon of 5.06 percent. The other...