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 R_n(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.