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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 me mention at the outset that this question is a revised version of this deleted MO question.

Let us first consider a certain special case of the definition in the following,

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

$$g(x):=\begin{cases}\dfrac{f(x)-f(a)}{x-a}&\text{if}~x\ne a\\L&\text{else} \end{cases}$$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. Let $g_a:X\to Y\times Y\times X\times X$ be the function defined by, $$g_a(x):=(f(x),f(a),x,a)$$for all $x\in X$. A function $f:X\to Y$ is said to be differentiable at $a\in X$ with value in a topological space $Z_a$ with respect to a function $$\psi_a:f(X)\times \{f(a)\}\times X\setminus\{a\}\times \{a\} \to Z_a$$ iff there exist a unique continuous function $$\Phi_a:f(X)\times \{f(a)\}\times X\times \{a\} \to Z_a$$such that $\Phi_a\circ g_a$ is continuous at $a\in X$ and $$\Phi_a\Big|_{f(X)\times \{f(a)\}\times X\setminus\{a\}\times \{a\}}\equiv \psi_a$$

For example, if $X=Y=Z_a=\mathbb{R}$ (equipped with usual topology) and $f$ is differentiable at $a\in \mathbb{R}$ with, $\psi_a: f(X)\times \{f(a)\}\times X\setminus\{a\}\times \{a\} \to Z_a$ defined as follows,

$$\psi_a\Bigl(f(x),f(a),x,a\Bigr)=\dfrac{f(x)-f(a)}{x-a}$$ We may then define $\Phi_a: f(X)\times \{f(a)\}\times X\times \{a\} \to Z_a$ as follows,

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

Some Remarks

  • It was pointed out to me by Alexander Schmending in a comment below the deleted MO post which I have mentioned earlier that this paper is related to my query although he explicitly mentioned that the framework of the paper is restrictive than what I want in this post. I have briefly read the paper and indeed found the approach of the paper to be based on the same idea of generalizing Caratheodory's definition of derivative to more general setting. However even after reading the paper I found no obvious way of generalizing the approach of the authors to arbitrary topological spaces.

  • Regarding the specific nature of relationship between $f$ and $\Phi_a\circ g_a$, what I have in mind is something like the following: if $f$ is differentiable at $a$ then $(\Phi_a\circ g_a)(a)$ denotes the value of a derivative of $f$ at $a$ with value in $Z_a$. (If we want a derivative of $f$ at $a$ we may impose the restriction that $Z_a$ be Hausdorff.)

Question

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

  • How should I go about formulating a notion of derivative of $f$ at $a$ with value in $Z_a$?

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  • $\begingroup$ Is there a Caratheodory's definition of derivative in higher dimension (say, for function $f : \mathbb R^2 \to \mathbb R$)? $\endgroup$ – Arctic Char Jun 3 at 7:36
  • $\begingroup$ Related $\endgroup$ – Arctic Char Jun 3 at 7:38
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    $\begingroup$ In Definition 1, you work on a division ring. Noting that a division ring is not necessarily commutative, then the fraction $\frac{f(x)-f(a)}{x-a}$ is ambiguous. You may write it like $(x-a)^{-1}(f(x)-f(a))$ or $(f(x)-f(a))(x-a)^{-1}$ as left or right fractions. $\endgroup$ – Qurultay Jun 3 at 7:39
  • $\begingroup$ @Qurultay: Thanks for pointing it out. $\endgroup$ – user 170039 Jun 3 at 13:33
  • $\begingroup$ @ArcticChar: See Definition 3 of this post. $\endgroup$ – user 170039 Jun 3 at 14:11

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