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The manifold has a priori no "smooth structure", i.e., there's no definition of derivatives there. The aim in the theory of differentiable manifolds is to define such an idea, and the construction principle is to map the set of points, making up a Hausdorff topological space, to the ##\mathbb{R}^d## and then via these maps you are able to define differentiation for functions defined on open subsets of the manifold in terms of differentiation of functions defined on open subsets of the ##\mathbb{R}^d## (with the standard topology). To make this consistent you need these constructions of charts and atlasses and all that.cianfa72 said:That was basically my point: we need a smooth structure for the manifold in the first place; only then you can check if a given function defined on the manifold (e.g. a coordinate function) is indeed a differentiable smooth function or is not.