# Analysis calculus proof kick start question

1. Mar 24, 2008

### gaborfk

1. The problem statement, all variables and given/known data
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 continous at all $$x\in\mathbb{R}$$.

2. Relevant equations

None

3. 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 continous also.

Last edited: Mar 24, 2008
2. Mar 24, 2008

### 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

3. Mar 24, 2008

### 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

4. Mar 25, 2008

### HallsofIvy

Staff Emeritus
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
$\lim_{h\rightarrow 0}f(x+a)= f(a)$.

5. Mar 25, 2008

### tiny-tim

Hi gaborfk!

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