Suppose $f$ is a continuous function of infinitely many real variables, and that 0 is an "identity element" for $f$ in the sense that

$$ f(0,\alpha,\beta,\gamma,\dots) = f(\alpha,\beta,\gamma,\dots). $$

Has anyone thought about the following limit (in particular, is there anything in the literature on it)?

$$ \lim_{\Delta\alpha\to0}\frac{f(\Delta\alpha,\ \alpha,\ \beta,\ \gamma,\dots) - f(\alpha+\Delta\alpha,\ \beta,\ \gamma,\ \dots) }{\Delta\alpha} $$

In case it makes anyone feel any better, for my purposes it may suffice to assume all but finitely many of the variable are zero (but of course with no prior finite bound on how many nonzero ones there are).