Let $f$ be a function from $\mathbb{R}^2$ to $\mathbb{R}$. If $f$ is separable in its coordinates, i.e $f(x,y) = g(x)h(y)$, does it satisfy some smoothness property? For example, is it necessary continuous on $\mathbb{R}^2$? With first derivative (if it exists) continuous on $\mathbb{R}^2$?
If $f$ is continuous on $\mathbb{R}^2$, it is not necessarily separable in its coordinates. Are there bounds known such that it can be approximately written with a sum of separable functions $g_i,h_i$ ?