A function on (some open subset of) a cartesian space with values in the real line is smooth, or infinitely differentiable, if all its derivatives exist at all points. More generally, if is any subset, a function is defined to be smooth if it has a smooth extension to an open subset containing .
For , a smooth map is a function such that is a smooth function for every linear functional . (In the case of finite-dimensional codomains as here, it suffices to take the to range over the coordinate projections.)
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).
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 local structures
|geometry||point||first order infinitesimal||formal = arbitrary order infinitesimal||local = stalkwise||finite|
|smooth functions||derivative||Taylor series||germ||smooth function|
|curve (path)||tangent vector||jet||germ of curve||curve|
|smooth space||infinitesimal neighbourhood||formal neighbourhood||open neighbourhood|
|function algebra||square-0 ring extension||nilpotent ring extension/formal completion||ring extension|
|arithmetic geometry||finite field||p-adic integers||localization at (p)||integers|
|Lie theory||Lie algebra||formal group||local Lie group||Lie group|
|symplectic geometry||Poisson manifold||formal deformation quantization||local strict deformation quantization||strict deformation quantization|