Flat function

In mathematics, especially real analysis, a flat function is a smooth function all of whose derivatives vanish at a given point . The flat functions are, in some sense, the antitheses of the analytic functions. An analytic function is given by a convergent power series close to some point :
In the case of a flat function we see that all derivatives vanish at , i.e. for all . This means that a meaningful Taylor series expansion in a neighbourhood of "x"0 is impossible. In the language of Taylor's theorem, the non-constant part of the function always lies in the remainder for all .
The function need not be flat at just one point. Trivially, constant functions on R are flat everywhere. But there are other, less trivial, examples.
Example.
The function defined by
is flat at . Thus, this is an example of a non-analytic smooth function.