In "Representation of functions of two variables as sums of rectangular functions, I", Roy O. Davies shows that under the continuum hypothesis every function has a representation of this form.

More precisely, he shows that we can get a representation with the additional property that for all $x,y$, the sum has only finitely many non-zero terms. Conversely, he proves that if there is a representation with this additional property for $(x,y) \mapsto e^{xy}$, then the continuum hypothesis holds.

In his survey Set Theoretic Real Analysis(p. 18), Krzysztof Ciesielski mentions this problem (and a lot of interesting similar results), it seems that it is open whether or not the continuum hypothesis is equivalent to the existence of weak representations.

In "Representation of functions of two variables as sums of rectangular functions, I", Roy O. Davies shows that under the continuum hypothesis every function has a representation of this form.

More precisely, he shows that we can get a representation with the additional property that for all $x,y$, the sum has only finitely many non-zero terms. Conversely, he proves that if there is a representation with this additional property for $(x,y) \mapsto e^{xy}$, then the continuum hypothesis holds.

In his survey Set Theoretic Real Analysis(p. 18), Krzysztof Ciesielski mentions this problem (and a lot of interesting similar results), it seems that it is open whether or not the continuum hypothesis is equivalent to the existence of weak representations. It is not open, cf. Santi Spadaro's comment below, or Péter Komjáth's answer.

In "Representation of functions of two variables as sums of rectangular functions, I", Roy O. Davies shows that under the continuum hypothesis every function has a representation of this form.

In "Representation of functions of two variables as sums of rectangular functions, I", Roy O. Davies shows that under the continuum hypothesis every function has a representation of this form.

More precisely, he shows that we can get a representation with the additional property that for all $x,y$, the sum has only finitely many non-zero terms. Conversely, he proves that if there is a representation with this additional property for $(x,y) \mapsto e^{xy}$, then the continuum hypothesis holds.

In his survey Set Theoretic Real Analysis(p. 18), Krzysztof Ciesielski mentions this problem (and a lot of interesting similar results), it seems that it is open whether or not the continuum hypothesis is equivalent to the existence of weak representations.