$f:V \to \mathbb C$ holomorphic , then $g(x,y):=\frac{f(x)-f(y)}{x-y} $ , $x \ne y , x,y \in V$ and $g(x,x)=f'(x) , \forall x \in V$ is continuous?

Question

Let $V\subseteq \mathbb C$ be an open connected set, $f:V \to \mathbb C$ be a holomorphic function. Then is the function $g:V \times V \to \mathbb C$ defined as $g(x,y):=\dfrac{f(x)-f(y)}{x-y}$, for $x \ne y , x,y \in V$ and $g(x,x)=f'(x), \forall x \in V$ continuous? I can prove that for $(x,y) \in V \times V$ with $x \ne y$, $(x,y)$ is a continuity point of $g$, but I am having trouble showing the continuity of $g$ on diagonal points. Please help. Thanks in advance

Answer 1

Your function is in fact holomorphic in the two variables. You may see this by writing: $$ g(x,y) = \int_0^1 f'(y+t(x-y)) \; dt $$ valid whenever the segment $[y;x]=\{tx + (1-t)y : 0\leq t \leq 1\} \subset V$. The integrand is (locally) holomorphic in both $x$ and $y$.