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• Flat_function abstract "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.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.".
• Flat_function thumbnail FBN_exp(-1x2).jpeg?width=300.
• Flat_function wikiPageID "27005472".
• Flat_function wikiPageRevisionID "550346578".
• Flat_function hasPhotoCollection Flat_function.
• Flat_function subject Category:Algebraic_geometry.
• Flat_function subject Category:Differential_calculus.
• Flat_function subject Category:Differential_structures.
• Flat_function subject Category:Real_analysis.
• Flat_function subject Category:Smooth_functions.
• Flat_function type Artifact100021939.
• Flat_function type DifferentialStructures.
• Flat_function type Object100002684.
• Flat_function type PhysicalEntity100001930.
• Flat_function type Structure104341686.
• Flat_function type Whole100003553.
• Flat_function type YagoGeoEntity.
• Flat_function type YagoPermanentlyLocatedEntity.
• Flat_function comment "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.".
• Flat_function label "Flat function".
• Flat_function sameAs m.0bs5xtg.
• Flat_function sameAs Q5457836.
• Flat_function sameAs Q5457836.
• Flat_function sameAs Flat_function.
• Flat_function wasDerivedFrom Flat_function?oldid=550346578.
• Flat_function depiction FBN_exp(-1x2).jpeg.
• Flat_function isPrimaryTopicOf Flat_function.