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# Cauhy-RiemannTheCauchy-Riemann equations and analyticity

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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

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

Question: Does it follow that $f$ is analytic everywhere in $D$?

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).The latter condition appears first in an earlier work by Besicovitch who showed that if a function defined on a domain $D$ is continuous everywhere and is $\mathbb{C}$-differentiable except on a countable union of closed sets of finite linear measure, it is analytic everywhere in $D$.

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# Cauhy-Riemann equations and analyticity

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

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

Question: Does it follow that $f$ is analytic everywhere in $D$?

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). The latter condition appears first in an earlier work by Besicovitch who showed that if a function defined on a domain $D$ is continuous everywhere and is $\mathbb{C}$-differentiable except on a countable union of closed sets of finite linear measure, it is analytic everywhere in $D$.