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.