I would be interested to learn if the following generalization of the classical Looman-Menchoff theorem is true.

Assume that the function $f=u+iv$, defined on a domain $D\subset\mathbb{C}$, is such that

- $u_x$, $u_y$, $v_x$, $v_y$ exist almost everywhere in $D$.
- $u$, $v$ satisfy the Cauchy–Riemann equations almost everywhere in $D$.
- $f=f(x,y)$ is separately continuous (in $x$ and $y$) in $D$.
- $f$ is locally integrable.

Question:Does it follow that $f$ is analyticeverywherein $D$?

**Remark 1.** Condition 3 is essential (take $f=1/z$).

**Remark 2.** G. Sindalovskiĭ proved analyticity of $f$ under conditions 2-4 when the partial derivatives exist everywhere in $D$, *except on a countable union of closed sets of finite linear Hausdorff measure* (link).