I'm looking for 3 analytic, continuous functions $f,g$ and $h$, all mapping $\mathbb{R}\to\mathbb{R}$, such that $$f(x+y)=g(x)+h(y)$$ On one hand, I think; this is not possible, as it fights the property of linearity. But on the otherhand, since $g$ and $h$ are not the same as $f$ (perse), the linearity property maybe is not ruining the fun here...
Does one of you maybe know an example? (or show that this is not possible)
I found some relating questions, however, these asked for a function $f$ that satisfies the property $f(x+y)=f(x)+f(y)$, which I don't require
You have :
$$ f(x+y)=g(x)+h(y) \tag{1} $$
Taking $y=0$ in (1) above, we deduce
$$ g(x)=f(x)-h(0) \tag{2} $$
Plugging back (2) into (1), we find
$$ f(x+y)=f(x)+h(y)-h(0) \tag{3} $$
Taking $x=0$ in (3) above, we have
$$ h(y)=f(y)-f(0)+h(0) \tag{4} $$
Plugging back (4) into (3), we find
$$ f(x+y)=f(x)+f(y)-f(0) \tag{5} $$
If we put $F(x)=f(x)-f(0)$, we can rewrite (5) as
$$ F(x+y)=F(x)+F(y) \tag{6} $$
Since $F$ is continuous, $F$ is linear and $f$ is affine. Note that you don't need analyticity, you just need continuity.