An example of a flat function at 0 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
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.