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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 ''Rn''(''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. Thus, this is an example of a non-analytic smooth function. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「flat function」の詳細全文を読む スポンサード リンク
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