Looman-Menchoff theorem

Statement

Verbal statement

If a function on an open subset of the complex numbers is continuous at every point and satisfies Cauchy-Riemann differential equations at every point, then it is holomorphic.

Symbolic statement

Let $U$ be a domain in the complex numbers, and $f:U\to \mathbb{C}$ be a continuous function, such that:

• For any point $z\in U$, the real and imaginary parts of $f$ have well-defined partial derivatives in the $x$ and $y$ directions
• These partial derivatives satisfy the Cauchy-Riemann differential equations

Then, $f$ is a holomorphic function.

Note: It is not true that if $f$ is continuous at a particular point $z\in U$ and has partial derivatives satisfying the Cauchy-Riemann differential equations, then it is complex-differentiable at that point.

Related facts

• Menchoff's theorem: A generalization, that only requires continuity and differentiability along two lines at every point, with the complex derivatives matching up.