Flat function explained

In mathematics, especially real analysis, a real function is flat at

x _{0} if all its derivatives at x _{0} exist and equal .

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 is the function such that

f( 0 ) =0 and $f(x)=e^$ for x ≠ 0 .

The function need not be flat at just one point. Trivially, constant functions on

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 $f(x)=e^$ for x> 0 .

Example

The function defined by

f(x) = \begin{cases} e ^{-1 /x ^{2}} & if x ≠ 0 \\ 0 & 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