Difference quotient of a complex-valued function

Definition

Suppoes $U\subset \mathbb{C}$ is an open subset and $f:U\to \mathbb{C}$ is a function. The difference quotient of $f$ is a function $F:(U\times U)\setminus (diag)\to \mathbb{C}$ given by:

$F(z,w):=\frac{f(z)-f(w)}{z-w}$

If $f$ is a holomorphic function, then $F$ can be extended to a continuous function on $U\times U$, by defining:

$F(z,z):=f^{\prime }(z)$

Facts

• Difference quotient of holomorphic function is holomorphic