I am interested in classifying solutions $f\,:\,\mathbb R\longrightarrow \mathbb R$ to the functional equation \begin{equation} f(x+y)=f(x)+f(y)+2xy\qquad\qquad(\dagger) \end{equation} and in particular, how to minimize the underlying assumptions of $f$. For instance, one can show that for every $a\in\mathbb R$, there exists a unique solution $f$ of $(\dagger)$ which is differentiable at zero with $f'(0)=a$. Indeed, it is a simple induction to verify that if $f$ is such a solution and $x\in\mathbb R$ then $$f(x)=2^nf(2^{-n}x)+x^2(1-2^{-n})\qquad\qquad(*)$$ for every $n\in\mathbb N$. (Those who are curious might like to derive this formula for themselves.) Since $2^nf(2^{-n})\to f'(0)=a$ as $n\to\infty$, it follows that $$f(x)=ax+x^2$$ which proves uniqueness, and existence follows simply by observing that $f$ as defined above is indeed a solution.
What if we drop the assumption that $f$ is differentiable at zero? It is simple to derive some basic properties of the solution, such as $f(0)=0$ and $f(x)+f(-x)=2x^2$. However, I cannot seem to get much more without referring to formula $(*)$ above, which is not that useful if we do not know a priori that $f'(0)$ exists.
Does anyone have any insights as to how to weaken the assumptions? Perhaps only continuity at zero, or even continuity everywhere?
