\frac{\Phi_{n-1}f(x_1,x_3,\dots,x_{n+1}) -
 \Phi_{n-1}f(x_2,\dots,x_{n+1})}{x_1-x_2}.$$

This definition does not depend on the choice of variables, as all difference quotients are symmetric functions. A function $f$ on $X$ is called a $C^n$ function if $\Phi_n f$ can be extended to a continuous function $\bar{\Phi}_{n}f : U^{n+1} \to X $. We also set $$D_n f(a) = \overline{\Phi_n}f(a,\dots,a),~a \in U.$$ We have the following theorem (c.f. [@S], Theorem $29.5$).

\[derivative\] Let $f \in C^{n}(U \to X)$. Then, $f$ is $n$ times differentiable and $$j!D_j f = f^j$$ for all $1 \leq j \leq n$.

To define $C^{k}$ functions in several variables, we follow the notation set forth in [@KT]. Consider a multiindex $\beta =
(i_1,\dots,i_d)$ and let $$\Phi_{\beta}f = \Phi^{i_1}_{1}\circ \dots \circ \Phi^{i_d}_{d} f.$$ This difference order quotient is defined on the set $
\nabla^{i_1}U_1 \times \dots \times \nabla^{i_d}U_d$ and the $U_i$ are all non-empty subsets of $X$ without isolated points. A function $f$ will then be said to belong to $C^{k}(U_1\times \dots
\times U_d)$ if for any multiindex $\beta$ with $|\beta| = \sum_{j =
1}^{d} i_j \leq k$, $\Phi_{\beta} f$ extends to a continuous function $\bar{\Phi}_{\beta}f : U_{1}^{i_1 + 1} \times \dots \times
U_{d}^{i_d + 1}$. We then have $$\label{multivanish}
\partial_{\beta}f(x_1,\dots,x_d) = \beta!
\bar{\Phi}_{\beta}(x_1,\dots,x_1,\dots,x_d,\dots,x_d)$$ 