I am currently watching a series of lectures which are a part of International Winter School on Gravity and Light 2015 (by Prof. Frederick P. Schuller). In one of the lectures on differentiable manifolds he asks- Is the structure of a topological manifold enought to talk about differentiability of curves on the manifold?Can you construct from it a notion of differentiability? And he answers in the negative saying that the structure of a topological manifold is not good enough to define a notion of differentiability.
Why is this true? What is missing in the structure of a topological manifold that is preventing us from defining differentiability on it?
Here is the video lecture link. Go to 01:43
If you could define a canonical differential structure in terms of topological data, then it would be homeomorphism-invariant, since homeomorphisms preserve topology. Once we have a differential structure, we know that locally we can talk about differentiability in terms of coordinates. The fact that there are local homeomorphisms of $\mathbb R^n$ that are not differentiable then leads to a contradiction.
For a concrete example, consider the topological manifold $M = \mathbb R$, the homeomorphism $\phi(x) = x + |x|/2$ thereof and the function $f : M \to \mathbb R : x \mapsto x$. This function $f$ is differentiable, but $f \circ \phi$ is not; so our differential structure is not homeomorphism-invariant. This example is pretty much universal: you can always find a homeomorphism that has the form $\psi(x,y,z,\ldots) = (\phi(x),y,z,\ldots)$ in some local coordinates.