Editing Smoothness (section)

===Relation to analyticity===
While all [[analytic function]]s are "smooth" (i.e. have all derivatives continuous) on the set on which they are analytic, examples such as [[bump function]]s (mentioned above) show that the converse is not true for functions on the reals: there exist smooth real functions that are not analytic. Simple examples of functions that are [[Non-analytic smooth function#A smooth function which is nowhere real analytic|smooth but not analytic at any point]] can be made by means of [[Fourier series]]; another example is the [[Fabius function]]. Although it might seem that such functions are the exception rather than the rule, it turns out that the analytic functions are scattered very thinly among the smooth ones; more rigorously, the analytic functions form a [[Meagre set|meagre]] subset of the smooth functions. Furthermore, for every open subset ''A'' of the real line, there exist smooth functions that are analytic on ''A'' and nowhere else.{{citation needed|date=December 2020}}

It is useful to compare the situation to that of the ubiquity of [[transcendental number]]s on the real line. Both on the real line and the set of smooth functions, the examples we come up with at first thought (algebraic/rational numbers and analytic functions) are far better behaved than the majority of cases: the transcendental numbers and nowhere analytic functions have full measure (their complements are meagre).

The situation thus described is in marked contrast to complex differentiable functions. If a complex function is differentiable just once on an open set, it is both infinitely differentiable and analytic on that set.{{citation needed|date=December 2020}}