A function  $f:A\times B\to C$ corresponds to the function $\tilde f:A\to C^B$ in the "typographical" isomorphism  $C^{A\times B}\sim (C^B)^A$ defined by $\tilde f(a)(b):=f(a,b)$ for all $a$ and $b$. If $A$ and $B$ are vector spaces on the field $C$, $ f$ writes in the form $f(a,b)=\sum_{i=1}^r u_i(a)v_i(b)$, that is $\tilde f(a)=\sum_{i=1}^r u_i(a)v_i$, if and only if $\tilde f$ takes values into an $r$-dimensional linear subspace $V\subset C^B$.

In your example (with $A=B$ a Hilbert space) the family of functions $\{b\mapsto \operatorname{sgn}(a\cdot b)\}_{a\in A}$ spans an infinite dimensional subspace of $C^A$. Indeed, for instance, if $S\subset A$ is such that no elements of $S$ are collinear (e.g. a hemisphere) then $\{b\mapsto \operatorname{sgn}(a\cdot b)\}_{a\in S}$ is a linearly independent family. The latter fact is apparent looking at the discontinuity set of these functions, as you were saying.