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).



A derivative of sorts?
I think Willie Wong's answer here suffices. This is just a case of looking at something from a suddenly different point of view and missing the obvious since it's not the way I'd been looking at it before.


I had an occasion to think about the difference
$$
f(\alpha,\beta,\gamma,\delta,\dots) - f(\alpha+\beta,\gamma,\delta,\dots)
$$
and was just playing around with that.


Continuity in the first variable is enough for the present purpose; I mentioned it only to rule out an obvious reason why the limit might not exist.


The functions I was thinking about were somewhat similar to the sum of products I mentioned in one of my comments above.


