A definition of differentiable functions for arbitrary topological spaces


Background

It is well-known that there is no notion of derivative for arbitrary topological spaces. However while investigating the notion of derivative as we find in one variable real analysis I came to generalize the notion. I am now wondering what properties must this notion of “differentiable function” satisfy in order to become a “good” enough definition.

Our motivation for the formulation of this definition is mainly Caratheodory’s definition of derivative.

The Definition

Before going straight into the definition itself, let us consider a certain special case of the definition in the following,

Definition 1. Let $ R$ be ring and $ \tau_1,\tau_2$ be any two topologies on $ R$ . A continuous function $ f:(R,\tau_1)\to (R,\tau_1)$ will then said to be $ (\tau_1,\tau_2)$ -differentiable on $ R$ at $ a\in R$ iff there exists a function $ g:(R,\tau_1)\to (R,\tau_1)$ such that,

$ $ f(x)-f(a)=g(x)(x-a)$ $ for all $ x\in R$ and $ g$ is continuous at $ a$ .

We can generalize the above definition as follows,

Definition 2. Let $ X,Y$ be arbitrary topological spaces. A continuous function $ f:X\to Y$ is said to be differentiable with respect to a function $ g:(f(X))^2\times X^2 \to Y$ at $ a\in X$ iff for any net $ (x_\alpha)_{\alpha\in J}$ converging to $ a$ , $ g$ is continuous at $ ((f(a),f(a)),(a,a))$ .

For example, if $ X=Y=\mathbb{R}$ (equipped with usual topology) and $ f$ is differentiable at $ a\in \mathbb{R}$ then we may define, $ g:(f(X))^2\times X^2 \to Y$ as follows,

$ $ g\Bigl((f(x),f(a)),(x,a)\Bigr)=\begin{cases}\dfrac{f(x)-f(a)}{x-a} &\text{if}~x\ne a\ f'(a)&\text{else}\end{cases}$ $

Also if $ X=U$ (open in $ \mathbb{R}^m$ ), $ Y=\mathbb{R}^n$ and $ f$ is differentiable at $ a\in U$ then we may define, $ g:(f(X))^2\times X^2 \to Y$ as follows,

$ $ g\Bigl((f(x),f(a)),(x,a)\Bigr)=\begin{cases}\dfrac{f(x)-f(a)-f'(x-a)}{\lVert x-a\rVert} &\text{if}~x\ne a\ 0&\text{else}\end{cases}$ $

Question

What properties should this notion of differentiable function must have so that it is a “good” enough definition of differentiable functions?