smooth map (Rev #3) in nLab

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