Editing Non-analytic smooth function (section)

==Complex analysis==
This pathology cannot occur with differentiable [[complex analysis|functions of a complex variable]] rather than of a real variable. Indeed, all [[holomorphic functions are analytic]], so that the failure of the function ''f'' defined in this article to be analytic in spite of its being infinitely differentiable is an indication of one of the most dramatic differences between real-variable and complex-variable analysis.

Note that although the function ''f'' has derivatives of all orders over the real line, the [[analytic continuation]] of ''f'' from the positive half-line ''x''&nbsp;>&nbsp;0 to the [[complex plane]], that is, the function

:<math>\mathbb{C}\setminus\{0\}\ni z\mapsto e^{-\frac{1}{z}}\in\mathbb{C},</math>

has an [[essential singularity]] at the origin, and hence is not even continuous, much less analytic. By the [[great Picard theorem]], it attains every complex value (with the exception of zero) infinitely many times in every neighbourhood of the origin.