Flat function

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

In mathematics, especially real analysis, a real function is flat at $x_0$ if all its derivatives at $x_0$ exist and equal 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 $f(x)=e^{-1/x^2}$ for $x\ne 0.$ 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 $x\le 0$ and $f(x)=e^{-1/x^2}$ for $x>0.$

Example

The function defined by

$f(x)=\{\begin{array}{ll}{e}^{-1/x^2}& \text{if }x\ne 0\\ 0& \text{if }x=0\end{array}$

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