# A derivative of sorts?

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).

• Could you elaborate on the setting you are interested in? Is there a reason you don't just take the usual partial derivative of $f$ with respect to $\alpha$ and keep the other variables fixed? Do you have an example $f$ in mind? It is probably important to pin down whether you are considering only finitely many nonzero variables or not and then to put a topology on the domain of $f$ for this to make any sense. May 25 '10 at 19:54
• Why would anyone want to consider such a limit? May 25 '10 at 19:55
• Being a bit naive here: isn't the limit you wrote equal to $\lim \frac{f(\triangle \alpha,\alpha,\beta,\ldots) - f(0,\alpha,\beta,\ldots)}{\triangle \alpha} - \frac{f(\alpha + \triangle\alpha,\beta,\ldots) - f(\alpha,\beta,\ldots)}{\triangle \alpha}$ using that $f$ is right-translation invariant? Which formally makes it just the difference of the two derivatives. Also, what is the topology you are using on "$\mathbb{R}^\infty$"? I.e. what do you mean by continuous? Just separately continuous in each of the variables? May 25 '10 at 20:13
• Applied to the function $$g_k(\theta_1,\theta_2,\theta_3,\dots) = \sum_{|A|=k}\prod_{i\in A}\sin\theta_i\prod_{i\not\in A}\cos\theta_i$$ this seems to yield (but correct me if I'm wrong) $$-(\cos\theta_1) g_{k-1}(\theta_2,\theta_3,\dots) - (\sin\theta_1)g_k(\theta_2,\theta_3,\dots) - \frac{\partial}{\partial\theta_1} g_k(\theta_1,\theta_2,\theta_3,\dots).$$ May 25 '10 at 21:44

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.