Analysis calculus proof kick start question

gaborfk

1. The problem statement, all variables and given/known data
Prove: If [tex]f[/tex] is defined on [tex]\mathbb{R}[/tex] and continuous at [tex]x=0[/tex], and if [tex]f(x_{1}+x_{2})=f(x_{1})+f(x_{2})[/tex] [tex]\forall x_{1},x_{2} \in\mathbb{R}[/tex], then [tex]f[/tex] is continous at all [tex]x\in\mathbb{R}[/tex].


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.

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 [tex]f(x_{1}+x_{2})=f(x_{1})+f(x_{2})[/tex] 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
[itex]\lim_{h\rightarrow 0}f(x+a)= f(a)[/itex].

tiny-tim

Hi gaborfk!

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