Flat function

In mathematics, especially real analysis, a flat function is a smooth function ƒ : ℝ → ℝ all of whose derivatives vanish at a given point x₀ ∈ ℝ. The flat functions are, in some sense, the antitheses of the analytic functions. An analytic function ƒ : ℝ → ℝ is given by a convergent power series close to some point x₀ ∈ ℝ:

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

The function y = e^−1/x2 is flat at x = 0.

In the case of a flat function we see that all derivatives vanish at x₀ ∈ ℝ, i.e. ƒ^(k)(x₀) = 0 for all k ∈ ℕ. This means that a meaningful Taylor series expansion in a neighbourhood of x₀ 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 ∈ ℕ.

The function need not be flat at just one point. Trivially, constant functions on ℝ are flat everywhere. But there are 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 x = 0. Thus, this is an example of a non-analytic smooth function.

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