# 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?