Flat function

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

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

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

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

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 ${\displaystyle x=0}$. 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