Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
14. Suppose that a function f : R → R satisfies f(x+y) = f(x)+ f (y) for each x, y Є R. We already know that f(0) = 0, f is an odd function, and that f(nx) = nf(x) for all x = R and n ≥ Z. a) Prove that f(qx) = qf(x) for all x = R and q Є Q. b) Prove that f is continuous at 0 if and only if f is continuous on R. c) Prove that if f is continuous at 0, then there is an m Є R such that f(x) = mx for all x = R.

14. Suppose that a function f : R → R satisfies f(x+y) = f(x)+ f (y) for each x, y Є R. We already know that f(0) = 0, f is an odd function, and that f(nx) = nf(x) for all x = R and n ≥ Z. a) Prove that f(qx) = qf(x) for all x = R and q Є Q. b) Prove that f is continuous at 0 if and only if f is continuous on R. c) Prove that if f is continuous at 0, then there is an m Є R such that f(x) = mx for all x = R.

Advanced Engineering Mathematics
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
icon
Related questions
Question

Do part C only Don't use PMI only use given things. 

14. Suppose that a function f : R → R satisfies f(x+y) = f(x)+ f (y) for each x, y Є R. We already know that f(0) = 0, f is an odd function, and that f(nx) = nf(x) for all x = R and n ≥ Z. a) Prove that f(qx) = qf(x) for all x = R and q Є Q. b) Prove that f is continuous at 0 if and only if f is continuous on R. c) Prove that if f is continuous at 0, then there is an m Є R such that f(x) = mx for all x = R.
Transcribed Image Text:14. Suppose that a function f : R → R satisfies f(x+y) = f(x)+ f (y) for each x, y Є R. We already know that f(0) = 0, f is an odd function, and that f(nx) = nf(x) for all x = R and n ≥ Z. a) Prove that f(qx) = qf(x) for all x = R and q Є Q. b) Prove that f is continuous at 0 if and only if f is continuous on R. c) Prove that if f is continuous at 0, then there is an m Є R such that f(x) = mx for all x = R.
Expert Solution
steps

Step by step

Solved in 2 steps

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,
SEE MORE TEXTBOOKS