Can we define a derivative on the $p$-adic numbers?

Or on any complete valuated field, really. This question came to me when I was thinking about real and complex differentiability, which we can define in a very similar manner: Let $\mathbb F\in\{\mathbb R,\mathbb C\}$, and $V,W$ normed, finite dimensional vector spaces over $\mathbb F$. Also let $f:V\to W$. We call $f$ $\mathbb F$-differentiable at $x_0\in V$ if there is an $\mathbb F$-linear map $\mathrm Df_{x_0}:V\to W$ such that $$\lim_{x\to x_0}\frac{\Vert f(x)-f(x_0)-\mathrm Df_{x_0}(x-x_0)\Vert}{\Vert x-x_0\Vert}=0.$$ This gives us a very natural way to talk about the real or complex differentiability of a function $\mathbb C\to\mathbb C$, for instance. Just consider $\mathbb C$ a two-dimensional vector space over $\mathbb R$ or a one-dimensional vector space over $\mathbb C$.

I'm wondering wether this can be generalized in an interesting way. This is my idea: We would need a generalization of a norm to general fields, and of course we would still need to work in complete spaces. Norms can be generalized using a valuation: Let $\mathbb F$ be a field. A valuation is a map $\vert\cdot\vert:\mathbb F\to\mathbb R$ satisfying the following for all $a,b\in\mathbb F$:

1. $\vert a\vert\geq0$ and $\vert a\vert=0$ iff $a=0$.
2. $\vert ab\vert=\vert a\vert\vert b\vert$.
3. $\vert a+b\vert\leq\vert a\vert+\vert b\vert$.

Now let $V$ be a vector space over $\mathbb F$. A norm on $V$ is a map $\Vert \cdot\Vert:V\to\mathbb R$ satisfying the following for all $v,w\in V,~a\in \mathbb F$:

1. $\Vert v\Vert\geq0$ and $\Vert v\Vert=0$ iff $v=0$.
2. $\Vert av\Vert=\vert a\vert\Vert v\Vert$.
3. $\Vert v+w\Vert\leq\Vert v\Vert+\Vert w\Vert$.

A norm on $V$ induces a metric $\Vert x-y\Vert$. We can now consider finite-dimensional, normed vector spaces $V,W$ over $F$ which are complete with respect to the induced metric, and are free to define: A function $f:V\to W$ is $\mathbb F$-differentiable in $x_0\in V$ if there exists an $\mathbb F$-linear map $\mathrm Df_{x_0}:V\to W$ such that $$\lim_{x\to x_0}\frac{\Vert f(x)-f(x_0)-\mathrm Df_{x_0}(x-x_0)\Vert}{\Vert x-x_0\Vert}=0.$$

Now the question is wether such a definition is actually interesting. For instance, finite fields and finite-dimensional vector spaces over them will be discrete, so while they are complete, evaluating the above limit is still not possible. But I imagine infinite fields, like the $p$-adic numbers, complex $p$-adic numbers, or function fields, could make the definition a sensible one. Has such a thing been done before? Are there any obstacles to the well-definedness? Or something that would make the theory of such derivatives uninteresting?

Solution 1:

Plenty of this has been done, and part of it is still actively researched. A very good textbook resource on the state of the art of the theory in the 1980s are chapters 2 to 4 of W. H. Schikhof's Ultrametric Calculus.

In particular, this contains many examples where that naive notion behaves weirdly (like in the other answer), even for someone who is already used to ultrametric spaces a bit. It turns out that to imitate $C^1$-functions in the $p$-adic setting, it is more useful to look at those $f$ where the limit

$$f'(a) := \lim_{(x,y) \to (a,a)} \frac{f(x)-f(y)}{x-y}.$$

exists. This is called strict differentiability e.g. in the article Robert Israel refers to in the comments. This theory and appropriate generalisations, as well as relation to appropriate notions of (locally) analytic functions, possible integrals and antiderivatives, Mahler bases etc., is expanded quite a bit in this book.

Solution 2:

This is well-defined but does not always behave as expected. For example, see page 5 of these nice notes for a $p$-adic function whose derivative is zero but which isn't even locally constant. https://www2.math.ethz.ch/education/bachelor/seminars/hs2011/p-adic/report8.pdf

Solution 3:

There is a short part on derivatives in $\mathbb{Q}_p$ in Fernando Q.Gouvêa's "$p$-adic numbers".

There is also no reason to restrict to finite-dimensional $V,W$. You might be interested in the notion of Fréchet derivatives.