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:

$F(z,z):=f'(z)$

Facts

• Difference quotient of holomorphic function is holomorphic