Derivatives for non Real/Complex functions

Question

I recently started on my journey into Abstract Algebra and am interested in learning about derivatives in a generalised sense. By that I mean all of my current understanding (or the overwhelming majority) about derivatives is in terms of Real and Complex Valued functions. I'm curious about what restrictions must exists so that we can define a functional limits on a given ordered Field $F$. It would seem that it must be ordered and complete, but how does continuity/limits come into play for functions of the following form

$f:F \longrightarrow F$

I was wondering if anyone knew of any good papers/textbooks/etc that speak to this.

Thanks in Advance, David

Answer 1

We don't even need an ordered field.

Just take $\mathbb{C}$, a typical non-ordered field, which has a well-defined derivative.

But even a field might be too much. In terms of abstract algebra, a ring is usually enough. Then the notion of a derivative can be described using linear maps such that the Leibniz rule for products holds.

Another common generalization uses differential geometry of topological spaces.

This wikipedia article serves as a good starting point for more information on the Generalizations of the derivative.

Answer 2

For such a map $f: F \rightarrow F$, one can say that $f$ is differentiable at $a\in F$ if $\lim \limits_{h\to 0} \frac{f(a+h)-f(a)}{h}$ exists, where the limit at $a \in F$ of a function $g: F \rightarrow F$ is $l \in F$ if $\forall \varepsilon \in F^{>0}, \exists \eta \in F^{>0}, \forall x \in F, |x-a|<\eta \rightarrow |g(x)-l|<\varepsilon$.

So this is just a basic generalisation of usual derivation. Likewise, continuity can be generalized to ordered fields. Then elementary properties of derivatives, such as continuity, stability under sum, product, composition etc hold but most important properties of real derivation, those which rely on the LUB to derive existence of limits, extrema, fixed points and so on, fail.

-For instance, on $\mathbb{Q}$, the caracteristic function of $\{x \in \mathbb{Q} \ | \ x^2 <2\}$ is continuous and everywhere differentiable with derivative $0$, but is not constant. For connectedness reasons, every ordered field besides $\mathbb{R}$ enjoys this phenomenon.

-See here and there respectively for the failure of l'Hospital's rule and of the theorem of the limit of the derivative in this context.

-See Real Analysis in Reverse of James Propp for a compilation of many properties related to continuity which fail for other ordered fields than $\mathbb{R}$.