Editing Analytic function (section)

==Analyticity and differentiability==
As noted above, any analytic function (real or complex) is infinitely differentiable (also known as smooth, or <math>\mathcal{C}^{\infty}</math>). (Note that this differentiability is in the sense of real variables; compare complex derivatives below.) There exist smooth real functions that are not analytic: see [[non-analytic smooth function]]. In fact there are many such functions.

The situation is quite different when one considers complex analytic functions and complex derivatives. It can be proved that [[proof that holomorphic functions are analytic|any complex function differentiable (in the complex sense) in an open set is analytic]]. Consequently, in [[complex analysis]], the term ''analytic function'' is synonymous with ''[[holomorphic function]]''.