The function ${\displaystyle y(x\neq 0)=e^{-1/x^{2}},}$ ${\displaystyle y(0)=0}$ is flat at ${\displaystyle x=0}$.

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

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

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

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

Example

The function defined by

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

is flat at ${\displaystyle x=0}$. 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

• 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