Flat function

In mathematics, especially real analysis, a flat function is a smooth function ƒ : ℝ → ℝ all of whose derivatives vanish at a given point x0 ∈ ℝ. The flat functions are, in some sense, the antitheses of the analytic functions.  An analytic function ƒ : ℝ → ℝ is given by a convergent power series close to some point x0 ∈ ℝ:

In the case of a flat function we see that all derivatives vanish at x0 ∈ ℝ, i.e. ƒ(k)(x0) = 0 for all k ∈ ℕ. This means that a meaningful Taylor series expansion in a neighbourhood of x0 is impossible.  In the language of Taylor's theorem, the non-constant part of the function always lies in the remainder Rn(x) for all n ∈ ℕ.

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 x = 0.  Thus, this is an example of a non-analytic smooth function. 

References

 

Category:Real analysis
Category:Algebraic geometry
Category:Differential calculus
Category:Smooth functions
Category:Differential structures