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.