Flat function

Function whose all derivatives vanish at a point
The function y ( x ≠ 0 ) = e − 1 / x 2 , {\displaystyle y(x\neq 0)=e^{-1/x^{2}},} y ( 0 ) = 0 {\displaystyle y(0)=0} is flat at x = 0 {\displaystyle x=0} .

In mathematics, especially real analysis, a real function is flat at x 0 {\displaystyle x_{0}} if all its derivatives at x 0 {\displaystyle x_{0}} exist and equal 0.

A function that is flat at x 0 {\displaystyle x_{0}} is not analytic at x 0 {\displaystyle x_{0}} unless it is constant in a neighbourhood of x 0 {\displaystyle x_{0}} (since an analytic function must equals the sum of its Taylor series).

An example of a flat function at 0 is the function such that f ( 0 ) = 0 {\displaystyle f(0)=0} and f ( x ) = e − 1 / x 2 {\textstyle f(x)=e^{-1/x^{2}}} for x ≠ 0. {\displaystyle x\neq 0.}

The function need not be flat at just one point. Trivially, constant functions on R {\displaystyle \mathbb {R} } are flat everywhere. But there are also other, less trivial, examples; for example, the function such that f ( x ) = 0 {\displaystyle f(x)=0} for x ≤ 0 {\displaystyle x\leq 0} and f ( x ) = e − 1 / x 2 {\textstyle f(x)=e^{-1/x^{2}}} for x > 0. {\displaystyle x>0.}

Example

The function defined by

f ( x ) = { e − 1 / x 2 if  x ≠ 0 0 if  x = 0 {\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 {\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