Let's define function:
$$g:\mathbb C \ni x \rightarrow f[a_1, a_2,...,a_n, x] \in \mathbb C$$
for $f \in C^{n + 1}$
I want to check the differentiability of function $g$ depending on $a_1, a_2,...,a_n$.
My work so far
If we consider point $x_0 \in \mathbb C$, our function will be differentiable when exists limit $\lim_{x \rightarrow x_0}\frac{g(x) - g(x_0)}{x - x_0}$.
Let's rewrite this formula:
$$\lim_{x \rightarrow x_0} \frac{g(x) - g(x_0)}{x - x_0} = \lim_{x \rightarrow x_0}\frac{f[a_1, a_2,...,a_n, x] - f[a_1, a_2, ..., a_n, x_0]}{x - x_0}$$
Now, I can permutate expressions within divided difference:
$$f[a_1, a_2,...,a_n, x, x_0] = f[x_0, a_1,...,a_n, x]$$
to obtain:
$$f[a_1, a_2,..., a_n, x, x_0] = f[x_0, a_1,...,a_n, x] = \frac{f[a_1,...,a_n, x] - f[x_0,x_1,...,a_n]}{x - x_0}= $$
$$=\frac{f[a_1,a_2,...,a_n, x] - f[a_1,a_2,...,a_n,x_0]}{x - x_0}$$
So:
$$\lim_{x \rightarrow x_0} \frac{g(x) - g(x_0)}{x - x_0} = \lim_{x \rightarrow x_0}f[a_1,a_2,...,a_n,x, x_0] = f[a_1,a_2,...,a_n,x_0,x_0]$$
And I'm not sure here about the last equality. I read somewhere that it's true but only, when $a_i \neq a_j$ for $i \neq j$, but I don't know where does it come from. Could you please explain to me why?