Menchoff's theorem

Statement

Let $U\subset \mathbb{C}$ be an open subset. A function $f:U\to mathbbC$ is holomorphic function iff the following two conditions hold:

• $f$ is continuous
• For any point $z_0\in U$, there exist two lines $l,l^{\prime }$ such that the following two limits:

$\underset{z\in l,z\to z_0}{lim}\frac{f(z)-f(z_0)}{z-z_0},\underset{z\in l^{\prime },z\to z_0}{lim}\frac{f(z)-f(z_0)}{z-z_0}$

exist and are equal.

Related facts

• Looman-Menchoff theorem: This is a corollary of Menchoff's theorem, where the two lines at each point are chosen as the vertical and horizontal lines