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 $0$.

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 $$x\neq 0.$$

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 $f(x)=e^{-1/x^2}$ 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.