• Support PF! Buy your school textbooks, materials and every day products Here!

Prove Continuity of f(x+y) = f(x) + f(y)

  • #1
strahi

Homework Statement


f(x+y) = f(x) + f(y) for all x,y∈ℝ if f is continuous at a point a∈ℝ then prove that f is continuous for all b∈ℝ.

Homework Equations


lim{x->a}f(x) = f(a)

The Attempt at a Solution


I do not understand how to prove the continuity, does f(x) = f(a) or does f(x+y) = f(a)
 

Answers and Replies

  • #2
FactChecker
Science Advisor
Gold Member
5,581
2,059
Write down what you need to prove for the continuity at b. See if you can simplify it using the property of f that is given. Then see if you can rewrite it with a and use the continuity at a.
 
  • #3
strahi
if it is continuous at b then lim{x->b)f(x) = f(b) right?
 
  • #4
Ray Vickson
Science Advisor
Homework Helper
Dearly Missed
10,706
1,728
if it is continuous at b then lim{x->b)f(x) = f(b) right?
Yes, that is one way to say it; but it is not the only way, and maybe not even the most useful way in this particular problem.
 
  • #5
strahi
sequential characterization then, or the proper definition of a limit?
 
  • #6
FactChecker
Science Advisor
Gold Member
5,581
2,059
Put that in the ε, δ form of the definition of a limit.
 
  • #7
33,631
5,288

Homework Statement


f(x+y) = f(x) + f(y) for all x,y∈ℝ if f is continuous at a point a∈ℝ then prove that f is continuous for all b∈ℝ.
It would help us as readers and you for understanding, if you used some punctuation and clarifying words.
In this problem it's given that f(x + y) = f(x) + f(y). It is also assumed that f is continuous at a real number a. You need to show (prove) that f is continuous at an real number b.

strahi said:

Homework Equations


lim{x->a}f(x) = f(a)

The Attempt at a Solution


I do not understand how to prove the continuity, does f(x) = f(a) or does f(x+y) = f(a)
You can let x be a, your call.

Another way that continuity is defined is this: ##\lim_{h \to 0} f(a + h) = f(a)##.
 
  • Like
Likes FactChecker

Related Threads on Prove Continuity of f(x+y) = f(x) + f(y)

  • Last Post
Replies
8
Views
29K
Replies
1
Views
5K
Replies
2
Views
2K
  • Last Post
Replies
5
Views
1K
Replies
2
Views
5K
  • Last Post
Replies
6
Views
2K
Replies
18
Views
2K
Replies
8
Views
2K
Replies
1
Views
955
Top