The concept can be generalised from cartesian spaces to Banach spaces and some other infinite-dimensional spaces. There is a locale-based analogue suitable for constructive mathematics which is not described as a function of points but as a special case of a continuous map (in the localic sense).
A manifold whose transition functions are smooth functions is a smooth manifold. The category Diff is the category whose objects are smooth manifolds and whose morphisms are smooth functions betweeen them.
Yet more generally, the morphisms between generalised smooth spaces are smooth maps.
For functions between manifolds that fall short of full smoothness, see differentiable map.
Basic facts about smooth functions are
the Hadamard lemma
A crucial property of smooth functions, however, is that they contain also bump functions.
Examples of sequences of infinitesimal and local structures
|first order infinitesimal||formal = arbitrary order infinitesimal||local = stalkwise||finite|
|derivative||Taylor series||germ||smooth function|
|tangent vector||jet||germ of curve||curve|
|Lie algebra||formal group||local Lie group||Lie group|
|Poisson manifold||formal deformation quantization||local strict deformation quantization||strict deformation quantization|