I read in Chapter 1 of Lee's Introduction to Smooth Manifolds that
there's no way to define a purely topological property that would serve as a criterion for smoothness.
So, I tried to think about the meaning of this sentence and I couldn't really link topology to differentiation! I mean derivatives of functions are defined on open domains, and openness is a topological abstract concept. But how are the two related? I mean assume that I change the topology of the real line from the Euclidean metric to some other topology. For example, discrete topology or the topology generated by half-open intervals $[a,b)$. How will the notion of differentiation change then?
I assume we have to study functions defined on $\mathbb{R}$ to answer this. So, some examples of functions that are differentiable with respect to the Euclidean topology but fail to be differentiable in some other topology or vice versa are appreciated.
As far as I can guess what Lee might have meant, on the basic level, the reason is that smoothness properties are not preserved by homeomorphisms, and hence smoothness of a function is not a topological property.
There is no need to go into exotic topologies (where it doesn't a priori make sense to talk about smoothness${}^\dagger$). For example, the identity function on the real line is certainly smooth, but if you compose it with a non-smooth homeomorphism of the line (say, a piecewise linear strictly increasing function), you will get a non-smooth function.
On a somewhat deeper level, you can have two differential manifolds which are homeomorphic, but have incompatible differential structures, such as regular and exotic spheres.
($\dagger$ There are more robust notions of smoothness and manifolds than those of real manifolds. For example, given any local field (like the $p$-adics), you can pretty much rewrite the standard definitions of smoothness and a manifold and they work just fine. But I am not aware of any such notion which would work for the lower limit topology.)