Flat function

In mathematics, especially real analysis, a flat function is a smooth function $f:\mathbb {R} \to \mathbb {R}$ all of whose derivatives vanish at a given point $x_{0}\in \mathbb {R}$ . The flat functions are, in some sense, the antitheses of the analytic functions. An analytic function $f:\mathbb {R} \to \mathbb {R}$ is given by a convergent power series close to some point $x_{0}\in \mathbb {R}$ :

f(x)\sim \lim _{n\to \infty }\sum _{k=0}^{n}{\frac {f^{(k)}(x_{0})}{k!}}(x-x_{0})^{k}.

In the case of a flat function, all derivatives vanish at $x_{0}\in \mathbb {R}$ , i.e. $f^{(k)}(x_{0})=0$ for all $k\in \mathbb {N}$ . This means that a meaningful Taylor series expansion in a neighbourhood of $x_{0}$ is impossible. In the language of Taylor's theorem, the non-constant part of the function always lies in the remainder $R_{n}(x)$ for all $n\in \mathbb {N}$ .

The function need not be flat at just one point. Trivially, constant functions on $\mathbb {R}$ are flat everywhere. But there are also other, less trivial, examples.

Example[edit]

The function defined by

f(x)={\begin{cases}e^{-1/x^{2}}&{\text{if }}x\neq 0\\0&{\text{if }}x=0\end{cases}}

is flat at $x=0$ . Thus, this is an example of a non-analytic smooth function. The pathological nature of this example is partially illuminated by the fact that its extension to the complex numbers is, in fact, not differentiable.

References[edit]

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