Flat function

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

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

A function that is flat at $x_{0}$ is not analytic at $x_{0}$ unless it is constant in a neighbourhood of $x_{0}$ (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 $f(0)=0$ and ${\textstyle f(x)=e^{-1/x^{2}}}$ for $x\neq 0.$

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

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. 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