Analysis calculus proof kick start question

[gaborfk]

Homework Statement

Prove: If $$f$$ is defined on $$\mathbb{R}$$ and continuous at $$x=0$$, and if $$f(x_{1}+x_{2})=f(x_{1})+f(x_{2})$$ $$\forall x_{1},x_{2} \in\mathbb{R}$$, then $$f$$ is continuous at all $$x\in\mathbb{R}$$.

Homework Equations

None

The Attempt at a Solution

Need a pointer to get started. Cannot wrap my head around it. I understand that I need to prove that the sum of two continuous functions is continuous also.

 

[ircdan]

there is only one function here, f , it has the property that f(x + y) = f(x) + f(y) for all x, y


hint, show f(0) = 0

 

[gaborfk]

I know that the function is continuous at x=0. So how does showing it is continuous at zero help with showing the function with the property $$f(x_{1}+x_{2})=f(x_{1})+f(x_{2})$$ is continuous?

Thank you

 

[HallsofIvy]

f is continuous at x= a if and only if
[tex]\lim_{x\rightarrow a}f(x)= f(a)[/itex].

If h= x- a, then x= a+ h and h goes to 0 as x goes to a: that becomes
$\lim_{h\rightarrow 0}f(x+a)= f(a)$.

 

[tiny-tim]

Hi gaborfk!

Hint: f(a + epsilon) = f(a) + f(epsilon)