nLab
smooth map

Definition

A function on (some subset of) a cartesian space ${ℝ}^n$ with values in the real line $ℝ$ is smooth, or infinitely differentiable, if all its derivatives exist at all points.

By coinduction: A function $f:ℝ\to ℝ$ is smooth if 1. it’s derivative exists and 2. the derivative is itself a smooth function.

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.

Properties

Basic facts about smooth functions are

• the Hadamard lemma

• Borel's theorem

• the Tietze extension theorem

• the Steenrod-Wockel approximation theorem

Examples

Every analytic functions (for instance a holomorphic function) is also a smooth function.

A crucial property of smooth functions, however, is that they contain also bump functions.

Related concepts

Examples of sequences of infinitesimal and local structures

first order infinitesimal	$\subset$	formal = arbitrary order infinitesimal	$\subset$	local = stalkwise	$\subset$	finite
	$\leftarrow$ differentiation		integration $\to$
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