Flat function
In mathematics, especially real analysis, a real function is flat at if all its derivatives at exist and equal.
A function that is flat at is not analytic at unless it is constant in a neighbourhood of .
An example of a flat function at is the function such that and for
The function need not be flat at just one point. Trivially, constant functions on are flat everywhere. But there are also other, less trivial, examples; for example, the function such that for and for
Example
The function defined byis 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.