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

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

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FactChecker
Gold Member
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.

strahi
if it is continuous at b then lim{x->b)f(x) = f(b) right?

Ray Vickson
Homework Helper
Dearly Missed
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.

strahi
sequential characterization then, or the proper definition of a limit?

FactChecker
Gold Member
Put that in the ε, δ form of the definition of a limit.

Mark44
Mentor

## 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)##.

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