Flat function

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

exist and equal 0.

A function that is flat at

unless it is constant in a neighbourhood of

(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

The function need not be flat at just one point.

Trivially, constant functions on

But there are also other, less trivial, examples; for example, the function such that

The function defined by is flat at

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.

The function is flat at .