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)

 

[FactChecker]

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]

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]

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

 

[Mark44]

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.

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: $\lim_{h \to 0} f(a + h) = f(a)$.