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Let us first define some terms.

Definition of Pre-pseudometric

Let $X\ne\emptyset$ and a function $\varphi:X\times X\to\mathbb{R}$ will be called a pre-pseudometric on $X$ if,

  1. $x=y\implies \varphi(x,y)=0$

  2. $\varphi(x,y)=-\varphi(y,x)$ for all $x,y\in X$

  3. $\varphi(x,y)=\varphi(x,z)+\varphi(z,y)$ for all $x,y,z\in X$

Definition of Pre-pseudometric Space

Let $X\ne\emptyset$ and $\varphi$ is a pre-pseudometric on $X$. Then $\langle X,\varphi\rangle$ will be called the pre-pseudometric space $X$ with pre-pseudometric $\varphi$.

It may be of interest to note at this point that if $\varphi$ is a pre-pseudometric on a non-empty set $X$ then a pseudometric $d_\varphi$ can be obtained from $\varphi$ by defining $d_\varphi(x,y)=|\varphi(x,y)|$.

Conversely, if $d_\varphi$ be a pseudometric on $X$ then function $\varphi(x,y)=d_\varphi(a,x)-d_\varphi(a,y)$ is a pre-pseudometric on $X$ where $a\in X$ is a fixed element.

Definition of Derivative of a Function in a Pre-pseudometric Space

Let $f:\langle X,\varphi_X\rangle \to \langle Y,\varphi_Y\rangle$. We will say that $f$ is differentiable at $a\in X$ if for all sequence $(\sigma_n)_{n\in\mathbb{N}}$ where $\sigma_n:=\dfrac{\varphi_Y(f(x_n),f(a))}{\varphi_X(x_n,a)}$ with $x_n\in X\setminus \{a\}$ and $\varphi_X(x_n,a)\ne 0$ for all $n\in\mathbb{N}$ and for all $\varepsilon>0$ we have, $$\left|\dfrac{\varphi_Y(f(x_n),f(a))}{\varphi_X(x_n,a)}-L\right|<\varepsilon$$ for some $L\in \mathbb{R}$ and for all sufficiently large $n\in\mathbb{N}$.

If this happens then we will say that $f$ is differentiable in the system $\bigl(\langle X,\varphi_X\rangle, \langle Y,\varphi_Y\rangle\bigr)$ and the derivative of the function at $x=a$ exists. We will then denote this derivative by $f'(a)$ and say that $f'(a)=L$.

Definition of Relative Derivative of a Function in a Pseudometric Space

We will say that a function $f$ is relatively differentiable in a pseudometric space $(X,d_\varphi)$ if with respect to a pre-pseudometric $\varphi$ that can be obtained from $d_\varphi$ it is differentiable in the pre-pseudometric space $\langle X,\varphi\rangle$.

Definition of Absolute Derivative of a Function in a Pseudometric Space

We will say that a function $f$ is absolutely differentiable in a pseudometric space $(X,d_\varphi)$ if with respect to every pre-pseudometric $\varphi$ that can be obtained from $d_\varphi$ it is differentiable in the pre-pseudometric space $\langle X,\varphi\rangle$.

Since a metric space is always a pseudometric space, I think that we have defined differentiation in an arbitrary metric space.


Question

Can we define the notion of "differentiation" or the "derivative of a function" as I have defined above?

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