**Context**

It is well known that there is no notion of a derivative for **arbitrary** topological spaces. However, by examining the notion of derivative as we find it in a variable, the actual analysis, I managed to generalize the notion. I now wonder what properties this notion of "differentiable function" must satisfy to become a "good" definition.

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

**The definition**

Before entering directly into the definition itself, consider a particular case of the definition in the following:

**Definition 1.** Let $ R $ ring and $ tau_1, tau_2 $ to be two topologies on $ R $. A continuous function $ f: (R, tau_1) to (R, tau_2) $ will then be said to be $ ( tau_1, tau_2) $-differentiable on $ R $ at $ a in R $ if there is a function $ g: (R, tau_1) to (R, tau_2) $ such as,

$$ f (x) -f (a) = g (x) (x-a) $$for everyone $ x in R $ and $ g $ is continuous to $ a $.

We can generalize the definition above as follows,

**Definition 2.** Let $ X, Y $ to 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 $** if for any net $ (x_ alpha) _ { alpha in J} $ converging towards $ a $, $ g $ is continuous to $ ((f (a), f (a)), (a, a)) $.

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

$$ g Bigl ((f (x), f (a)), (x, a) Bigr) = begin {case} dfrac {f (x) -f (a)} {xa} & text {if} ~ x does not \ f (&) text {else} end {cases} $$

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

$$ g Bigl ((f (x), f (a)), (x, a) Bigr) = begin {case} dfrac {f (x) -f (a) -f & # 39; (xa)} { lVert xa rVert} & text {if} ~ x ne \ 0 & text {else} end {cases} $$

**Question**

What properties should this notion of differentiable function have to be a "good" definition of differentiable functions?