Hello!! Let $f : \mathbb{R}\rightarrow \mathbb{R}$ be a function with $ f(x + y) = f(x) + f(y)$ for all $x, y \in \mathbb{R}$.
I want to show that $f$ is continuous if and only if $f$ is continuous in $0$.
The direction $\Rightarrow$ is trivial.
For the direction $\Leftarrow$ : Suppose that $f$ is continuous at $0$ then $\lim_{x\rightarrow 0)f(x)=f(0)$ and from the property above we have that $f(0)=0$.
How could we continue? I got stuck right now...
Im assuming the identities below can be proven with telescoping techniques and / or differentiation.
For $x>0$ let $f(x) = \sqrt{x + {1\over2}} ,\; f(x,0) = x,\; f(x,n) = f( f(x,n-1) )$.
Now we get
$$ - f(x)^4 + f(x,2)^4 - f(x,3)^4 + f(x,4)^4 - ... = 1/8 - x^2 $$
Where the LHS uses Cesàro s...
