Does there exist a continuous $f:\Bbb R^2\to\Bbb R$ not expressible as a finite sum $f(x,y)=\sum_i g_i(x)h_i(y)$?

Question

This feels like it should be very trivially false, but I'm struggling to find some criterion that holds for all functions $\sum_i g_i(x)h_i(y)$ that does not hold for all continuous $\Bbb R^2\to\Bbb R$, and I haven't managed to construct any obvious counter example.

Answer 1

Given $f:\mathbb{R}^2\to\mathbb{R}$ and any $a\in\mathbb{R}$, there is a function $f_a:\mathbb{R}\to\mathbb{R}$ given by $f_a(y)=f(a,y)$. If $f(x,y)=\sum_i g_i(x)h_i(y)$, then $f_a$ is an $\mathbb{R}$-linear combination of the functions $h_i(y)$ for each $a$. So if you have any function $f$ such that infinitely many of the functions $f_a$ are linearly independent over $\mathbb{R}$, it cannot be written in this form. (In fact, if you are just talking about arbitrary functions and have no restrictions like continuity, the converse holds as well: if the span of the functions $f_a$ is finite-dimensional, it is possible to write $f(x,y)=\sum_i g_i(x)h_i(y)$. Just take the $h_i$ to be a basis for the span of the $f_a$, and let $g_i(a)$ be the coefficients of $f_a$ for this basis.)

For instance, take $f(x,y)=|y|^{x^2+1}$. Since a nonzero polynomial can only have finitely many roots, the functions $|y|^n$ are linearly independent when $n$ ranges over the natural numbers. It follows that the functions $f_a(y)$ are linearly independent when $a$ ranges over the natural numbers, so $f$ cannot be written in the form $\sum_i g_i(x)h_i(y)$.