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 $f : U \to C$ 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.