Flat function

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

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:

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 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

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

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