I think it is impossible, but why?

Why is the question closed, although no one can answer it? Using topological structure we can define continuous functions and convergent sequences. Why differentiable functions could not be defined in terms of the topology?

Ali Taghavi: What do you mean with complex valued continuous functions on $X$ and $Y$, whereas you have defined $X$ and $Y$ as topological spaces?

Joseph Van Name: Do you have any analogous random idea for $f,g : X \to Y$, where $X$ and $Y$ are topological spaces?

Joseph Van Name: You are right. I got confused because you have defined a function between a topological space and $\mathbb{R}$.

Joseph Van Name: you should define a metric on $X$ at first for you idea to be right or have a meaning. I am not sure. This like the answer of Todd Trimble, on metric space.

If you have any suggestion for improving the question, please post it!

What do you mean exactly with "goal"? Please give examples.

you can say that differential topology is dealing with differentiable functions on differentiable manifolds. My question is dealing with differentiable functions on topological space in general. Is there an analogy in general topological space without any added structure or restrictions?

en.wikipedia.org/wiki/Generalizations_of_the_derivative

I just mean the definition. Is there any definition of differentiability of maps between topological space.