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 Rn(x) for all n ∈ ℕ.

Notice that the function need not be flat everywhere. The constant functions on ℝ are flat functions at all of their points. But there are other, non-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

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