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The function $y(x\neq 0)=e^{-1/x^{2}},$ $y(0)=0$ 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 ${\textstyle f(x)=e^{-1/x^{2}}}$ 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