# 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)

Related Calculus and Beyond Homework Help News on Phys.org
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)##.

• FactChecker