# 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 *x*_{0} ∈ 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 *x*_{0} ∈ ℝ:

In the case of a flat function we see that all derivatives vanish at *x*_{0} ∈ ℝ, i.e. ƒ^{(k)}(*x*_{0}) = 0 for all *k* ∈ ℕ. 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 *R _{n}*(

*x*) for all

*n*∈ ℕ.

Notice that the function need not be flat everywhere. The constant functions on ℝ are flat functions at all of their points. But there are other, non-trivial, examples.

## Example

The function defined by

is flat at *x* = 0.

## References

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