Proving Continuity of Functions on the Reals

[Math_Geek]

Homework Statement

Prove: If f is defined on the reals and continuous at x=0, and if f(x1+x2)=f(x1)+f(x2) for all x1,x2 in the reals, then f is continuous at all x in the reals.

Homework Equations

Using defn of limits and continuity

The Attempt at a Solution

is this like proving that the sum of two functions is continuous? I am a bit confused, this is the last on of the homework, then I can enjoy my easter.

 

[morphism]

is this like proving that the sum of two functions is continuous? I am a bit confused, this is the last on of the homework, then I can enjoy my easter.

No. f is a function that satisfies the functional equation f(x_1 + x_2) = f(x_1) + f(x_2). You're told that f is continuous at 0, and you're supposed to use this to conclude that f is continuous everywhere.

 

[Math_Geek]

so show since f is continuous at 0, there exists an epsilion >0 such that and deal with the two functions?

 

[morphism]

There's just one function here: f. You want to prove that it's continuous (on all of R).

 

[Math_Geek]

ok so how does the f(x1)+f(x2) play into it?

 

[morphism]

You also know something else. f is continuous at 0.

Now use the definition of continuity and these two facts. You might also find it helpful to prove that f(-x) = -f(x) [hint: f(0)=0].

 

[tiny-tim]

ok so how does the f(x1)+f(x2) play into it?

Hint: what is f(x + epsilon) - f(x)?

 

[HallsofIvy]

$$\lim_{x\rightarrow a} f(x)= \lim_{h\rightarrow 0}f(a+ h)$$
where h= x- a. Then use the fact that f(a+ h)= f(a)+ f(h).