The $k$-th divided difference of a function $g$ at sites $\tau_i,\ldots, \tau_{i+k}$ is the leading coefficient (that is, the coefficient of $x^k$) of the polynomial of order $k+1$ that agrees with $g$ at the sequence $(\tau_i,\ldots, \tau_{i+k})$. It is denoted by $[\tau_i,\ldots, \tau_{i+k}]g$. In particular, if $g$ is itself a polynomial of order $k+1$, then $[\tau_i,\ldots, \tau_{i+k}]g$ is constant as a function of $\tau_i,\ldots, \tau_{i+k}$. If $g$ is a polynomial of order larger than equal to $k$, then $[\tau_i,\ldots, \tau_{i+k}]g=0$. As for specific values, $[\tau_1]g=g(\tau_1)$ and $$ [\tau_1, \tau_{2}]g=\frac{g(\tau_2)-g(\tau_1)}{\tau_2-\tau_1}, $$ provided that $\tau_1 \neq \tau_2$.
Also, if $g \in C^{k}$, i.e. if $g$ has $k$ continuous derivatives, then $\exists\zeta \in [\min(\tau_i, \tau_{i+k}), \max(\tau_i, \tau_{i+k})]$ so that $$ [\tau_i,\ldots, \tau_{i+k}]g=\frac{g^{(k)}(\zeta)}{k!}. $$ As a consequence, $$ [\tau_i,\ldots, \tau_{i+k}]g=\frac{g^{(k)}(\tau_i)}{k!}, \quad \tau_i=\ldots=\tau_{i+k} \text{ and } g\in C^k, $$ while $$ [\tau_i,\ldots, \tau_{i+k}]g =\frac{[\tau_i,\ldots,\tau_{r-1},\tau_{r+1},\ldots \tau_{i+k}]g-[\tau_i,\ldots,\tau_{s-1},\tau_{s+1},\ldots \tau_{i+k}]g}{\tau_s - \tau_r}, \quad \tau_r \neq \tau_s. $$
My question is the following: let $g\notin C^k$ and $\tau_i=\ldots=\tau_{i+k}$, then is the quantity $[\tau_i,\ldots, \tau_{i+k}]g$ defined in this case? If yes, what is it equal to?