Menchoff's theorem

Statement

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

• $f$ is continuous
• For any point $z_{0}\in U$, there exist two lines $l,l'$ such that the following two limits:

$\lim _{z\in l,z\to z_{0}}{\frac {f(z)-f(z_{0})}{z-z_{0}}}\qquad ,\qquad \lim _{z\in l',z\to z_{0}}{\frac {f(z)-f(z_{0})}{z-z_{0}}}$

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