Flat function

In mathematics, especially real analysis, a real function is flat at ${\displaystyle x_{0}}$ if all its derivatives at ${\displaystyle x_{0}}$ exist and equal 0.

A real function is constant in a neighbourhood of a point ${\displaystyle x_{0}}$ in the interior of its domain if and only if the function is flat at ${\displaystyle x_{0}}$ and is analytic at ${\displaystyle x_{0}}$.

An example of a function is flat at only at an isolated point is ${\displaystyle f:\mathbb {R} \to \mathbb {R} }$ such that ${\displaystyle f(0)=0}$ and that for all ${\displaystyle x\in \mathbb {R} }$, ${\displaystyle x\neq 0}$ implies ${\displaystyle f(x)=e^{-1/x^{2}}}$; the function ${\displaystyle f}$ is flat only at ${\displaystyle 0}$.

Since ${\displaystyle f}$ is not analytic at ${\displaystyle 0}$, the extension of ${\displaystyle f}$ to ${\displaystyle \mathbb {C} }$, that is the function ${\displaystyle f_{ext}:\mathbb {C} \to \mathbb {C} }$ such that ${\displaystyle f_{ext}(0)=0}$ and that for all ${\displaystyle z\in \mathbb {C} }$, ${\displaystyle z\neq 0}$ implies ${\displaystyle f_{ext}(z)=\exp(-z^{-2})}$, is not holomorphic at ${\displaystyle 0}$, due to that for complex functions, holomorphicity at a point implies analyticity at that point.

• Glaister, P. (December 1991), A Flat Function with Some Interesting Properties and an Application, The Mathematical Gazette, Vol. 75, No. 474, pp. 438–440, JSTOR 3618627