Question: For $z \in \mathbb{C}^{N+1}$ and $j \in \{1,\dots,N\}$, let $z(j) \in \mathbb{C}^N$ be the tuple obtained by omitting the $j^{\text{th}}$ coordinate of $z$. Let $D \subset \mathbb{C}^N$ be open and define $$D_j=\{z \in \mathbb{C}^{N+1}: z(j),z(j+1) \in D\}.$$ For any holomorphic function $f: D \rightarrow \mathbb{C}$ and each $j$, show that there exists a unique holomorphic function $\Delta_jf: D_j \rightarrow \mathbb{C}$ such that $$\Delta_jf=\frac{f(z(j+1))-f(z(j))}{z_j-z_{j+1}}$$ where $z_j\neq z_{j+1}$.
I don't know how to begin solving this, even in the simplest case with $N=1$ I become stuck. It looks suspiciously like some kind of mean value theorem although that may just be coincidence.
Any help is much appreciated!