Difference quotient of holomorphic function is holomorphic

Statement

Suppose $U\subset \mathbb {C}$ is an open subset and $f:U\to \mathbb {C}$ is a holomorphic function. Consider the function $F:U\times U\to \mathbb {C}$ given by:

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

Then, for any fixed value of $w\in \mathbb {C}$, the function:

$z\mapsto F(z,w)$

is holomorphic.