Flat function

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In real analysis, a real function is defined to be flat at a point in its domain if all its derivatives or partial derivatives exist at that point and equal ${\displaystyle 0}$ .

A real function is locally constant (that is, constant in at least one neighbourhood) of a point in the interior of its domain if and only if the function is flat and analytic at that point.

An example of a function that is flat only at an isolated point is ${\displaystyle f:\mathbb {R} \to \mathbb {R} }$ such that ${\displaystyle f(0)=0}$ and that for all ${\displaystyle x\in \mathbb {R} }$ , ${\displaystyle x\neq 0}$ implies ${\displaystyle f(x)=e^{-1/x^{2$ ; the function ${\displaystyle f}$ is flat only at ${\displaystyle 0}$ .

Since ${\displaystyle f}$ is not analytic at ${\displaystyle 0}$ , the extension of ${\displaystyle f}$ to ${\displaystyle \mathbb {C} }$ is not holomorphic at ${\displaystyle 0}$ , since for complex functions, holomorphicity at a point implies analyticity at that point.

Examples of construction of non-trivial flat functions#edit

By a non-trivial flat function, what is meant is a function that, at least at one point in the interior of its domain, is flat but not locally constant.

Construction of univariate flat functions#edit

Let ${\displaystyle a}$ be a positive real number and let ${\displaystyle g:S\to \mathbb {R} }$ (where ${\displaystyle S\subseteq \mathbb {R} }$ is a neighbourhood of a point ${\displaystyle x_{0}\in \mathbb {R} }$ ) be such that ${\displaystyle g(x_{0})=0}$ and that for all ${\displaystyle x\in S}$ , ${\displaystyle x\neq x_{0$ implies ${\displaystyle g(x)=\operatorname {exp} (-|x-x_{0}|^{-a}).}$

Then ${\displaystyle g}$ is flat at ${\displaystyle x_{0$ .

Construction of multivariate flat functions#edit

Let ${\displaystyle G:\mathbb {R} \to \mathbb {R} }$ be flat at ${\displaystyle 0}$ , and let ${\displaystyle H:P\to \mathbb {R} }$ (where ${\displaystyle n\in \mathbb {N} }$ , ${\displaystyle \mathbf {x} _{0$ is an ${\displaystyle n}$ -dimensional real coordinate vector, and ${\displaystyle P\subseteq \mathbb {R} ^{n$ is a neighbourhood of ${\displaystyle \mathbf {x} _{0$ ) be such that for all ${\displaystyle \mathbf {x} \in P}$ , ${\displaystyle H(\mathbf {x} )=G(||\mathbf {x} -\mathbf {x} _{0}||)}$ , where for all ${\displaystyle \mathbf {p} \in \mathbb {R} ^{n$ , ${\displaystyle ||\mathbf {p} ||}$ denotes the Euclidean norm of ${\displaystyle \mathbf {p} }$ .

Then ${\displaystyle H}$ is flat at ${\displaystyle \mathbf {x} _{0$ .

A necessary condition for flatness and local non-constancy#edit

Let ${\displaystyle S\subseteq \mathbb {R} ^{n$ for some ${\displaystyle n\in \mathbb {N} }$ and let ${\displaystyle F:S\to \mathbb {R} }$ be flat at a point ${\displaystyle x_{0$ in the interior of ${\displaystyle S}$ . Also let it be the case that for every neighbourhood ${\displaystyle N}$ of ${\displaystyle x_{0$ , there exists an ${\displaystyle x\in N}$ such that ${\displaystyle F(x)\neq F(x_{0})}$ , that is, that ${\displaystyle F}$ is not locally constant at ${\displaystyle x_{0$ . Then ${\displaystyle F}$ is non-analytic at ${\displaystyle x_{0$ .

Proof#edit

Assume the contrary, that is, that ${\displaystyle F}$ is analytic at ${\displaystyle x_{0$ . Since ${\displaystyle F}$ is flat at ${\displaystyle x_{0$ , the Taylor series of ${\displaystyle F}$ at ${\displaystyle x_{0$ is constant and equal to ${\displaystyle F(x_{0})}$ . Since it is assumed that ${\displaystyle F}$ is analytic at ${\displaystyle x_{0$ , then there exists a neighbourhood ${\displaystyle N}$ of ${\displaystyle x_{0$ such that for all ${\displaystyle x\in N}$ , ${\displaystyle F(x)=F(x_{0})}$ . This contradicts that for every neighbourhood ${\displaystyle N}$ of ${\displaystyle x_{0$ , there exists an ${\displaystyle x\in N}$ such that ${\displaystyle F(x)\neq F(x_{0})}$ . Hence, by contradiction, ${\displaystyle F}$ is non-analytic at ${\displaystyle x_{0$ .

A sufficient condition for flatness#edit

Let ${\displaystyle S\subseteq \mathbb {R} ^{n$ for some ${\displaystyle n\in \mathbb {N} }$ and let ${\displaystyle F:S\to \mathbb {R} }$ be infinitely differentiable at a point ${\displaystyle x_{0$ in the interior of ${\displaystyle S}$ . Also let it be the case that for every neighbourhood ${\displaystyle N}$ of ${\displaystyle x_{0$ , there exists an ${\displaystyle x\in N}$ such that ${\displaystyle F}$ is flat at ${\displaystyle x}$ . Then ${\displaystyle F}$ is flat at ${\displaystyle x_{0$ .

Proof#edit

Assume the contrary, that is, that ${\displaystyle F}$ is not flat at ${\displaystyle x_{0$ . Then there exists a ${\displaystyle k\in \mathbb {N} }$ such that a ${\displaystyle k}$ -th partial derivative of ${\displaystyle F}$ (call it ${\displaystyle F_{k$ ) is non-zero at ${\displaystyle x_{0$ , that is, ${\displaystyle F_{k}(x_{0})=r}$ for some ${\displaystyle r\in \mathbb {R} }$ such that ${\displaystyle r\neq 0}$ . Since ${\displaystyle F}$ is infinitely differentiable at ${\displaystyle x_{0$ , then ${\displaystyle F_{k$ is continuous at ${\displaystyle x_{0$ . Since ${\displaystyle r\neq 0}$ , then ${\displaystyle |r|/2>0}$ . Then there exists a neighbourhood ${\displaystyle N}$ of ${\displaystyle x_{0$ such that for all ${\displaystyle x\in N}$ , ${\displaystyle |F_{k}(x)-F_{k}(x_{0})|<|r|/2}$ , which means ${\displaystyle |F_{k}(x)-r|<|r|/2}$ , or, in other words, ${\displaystyle F_{k}(x)}$ lies in the open interval ${\displaystyle (\operatorname {min} \{r/2,3r/2\},\operatorname {max} \{r/2,3r/2\})}$ . Since ${\displaystyle r\neq 0}$ , ${\displaystyle 0\notin (\operatorname {min} \{r/2,3r/2\},\operatorname {max} \{r/2,3r/2\})}$ , so ${\displaystyle F_{k}(x)\neq 0}$ , which means that there exists a ${\displaystyle k\in \mathbb {N} }$ such that a ${\displaystyle k}$ -th partial derivative of ${\displaystyle F}$ is non-zero at ${\displaystyle x}$ . This contradicts that ${\displaystyle F}$ is flat at at least one point in every neighbourhood of ${\displaystyle x_{0$ . Hence, by contradiction, ${\displaystyle F}$ is flat at ${\displaystyle x_{0$ .

The above results can be used to show that a bump function is flat and non-analytic at each boundary point of the closure of its support.

Flatness of smooth interpolations#edit

Let ${\displaystyle s_{1}\in \mathbb {R} }$ and ${\displaystyle s_{2}\in \mathbb {R} }$ be such that ${\displaystyle s_{1}<s_{2$ .

Let ${\displaystyle I_{1}\subset \mathbb {R} }$ be an interval with non-empty interior, with supremum ${\displaystyle s_{1$ , and containing ${\displaystyle s_{1$ ; and let ${\displaystyle I_{2}\subset \mathbb {R} }$ be an interval with non-empty interior, with infimum ${\displaystyle s_{2$ , and containing ${\displaystyle s_{2$ .

In the following, continuity, one-sided continuity, one-sided limits, differentiability and smoothness of a real coordinate vector-valued function are respectively given by continuity, one-sided continuity, one-sided limits, differentiability and smoothness of the function in each coordinate.

Let ${\displaystyle n\in \mathbb {N} }$ . Let ${\displaystyle \mathbf {r} _{1}:I_{1}\to \mathbb {R} ^{n$ be continuously differentiable at every point in the interior of ${\displaystyle I_{1$ , left-continuous at ${\displaystyle s_{1$ and have the left-hand limit of its derivatives of all orders be finite at ${\displaystyle s_{1$ ; also let ${\displaystyle ||\mathbf {r} _{1}'(s)||=1}$ for all ${\displaystyle s\in \operatorname {int} (I_{1})}$ . Let ${\displaystyle \mathbf {r} _{2}:I_{2}\to \mathbb {R} ^{n$ be continuously differentiable at every point in the interior of ${\displaystyle I_{2$ , right-continuous at ${\displaystyle s_{2$ and have the right-hand limit of its derivatives of all orders be finite at ${\displaystyle s_{2$ ; also let ${\displaystyle ||\mathbf {r} _{2}'(s)||=1}$ for all ${\displaystyle s\in \operatorname {int} (I_{2})}$ .

Let curves ${\displaystyle C_{1$ and ${\displaystyle C_{2$ be the images of the domains of ${\displaystyle \mathbf {r} _{1$ and ${\displaystyle \mathbf {r} _{2$ , respectively. Both ${\displaystyle C_{1$ and ${\displaystyle C_{2$ inhabit ${\displaystyle \mathbb {R} ^{n$ .

A smooth interpolation between ${\displaystyle C_{1$ and ${\displaystyle C_{2$ , between the points ${\displaystyle \mathbf {r} _{1}(s_{1})}$ and ${\displaystyle \mathbf {r} _{2}(s_{2})}$ , is the image of the domain of a function ${\displaystyle \mathbf {r} _{0}:(s_{1},s_{2})\to \mathbb {R} ^{n$ such that the left-hand limit of ${\displaystyle \mathbf {r} _{0$ at ${\displaystyle s_{1$ is ${\displaystyle \mathbf {r} _{1}(s_{1})}$ , the right-hand limit of ${\displaystyle \mathbf {r} _{0$ at ${\displaystyle s_{2$ is ${\displaystyle \mathbf {r} _{2}(s_{2})}$ , and for all ${\displaystyle k\in \mathbb {N} }$ , the left-hand limit of the ${\displaystyle k}$ -th derivative of ${\displaystyle \mathbf {r} _{0$ at ${\displaystyle s_{1$ is equal to the right-hand limit of the ${\displaystyle k}$ -th derivative of ${\displaystyle \mathbf {r} _{1$ at ${\displaystyle s_{1$ , and the right-hand limit of the ${\displaystyle k}$ -th derivative of ${\displaystyle \mathbf {r} _{0$ at ${\displaystyle s_{2$ is equal to the left-hand limit of the ${\displaystyle k}$ -th derivative of ${\displaystyle \mathbf {r} _{2$ at ${\displaystyle s_{2$ . A smooth interpolation between ${\displaystyle C_{1$ and ${\displaystyle C_{2$ is defined to have ${\displaystyle G^{\infty$ continuity (geometric continuity of all orders) with ${\displaystyle C_{1$ and ${\displaystyle C_{2$ .

Let ${\displaystyle \mathbf {r} :I_{1}\cup (s_{1},s_{2})\cup I_{2}\to \mathbb {R} ^{n$ be such that: for all ${\displaystyle s\in I_{1$ , ${\displaystyle \mathbf {r} (s)=\mathbf {r} _{1}(s)}$ ; for all ${\displaystyle s\in (s_{1},s_{2})}$ , ${\displaystyle \mathbf {r} (s)=\mathbf {r} _{0}(s)}$ ; and for all ${\displaystyle s\in I_{2$ , ${\displaystyle \mathbf {r} (s)=\mathbf {r} _{2}(s)}$ .

If ${\displaystyle C_{1$ and ${\displaystyle C_{2$ are straight line segments, ${\displaystyle \mathbf {r} }$ is necessarily flat at ${\displaystyle s_{1$ and ${\displaystyle s_{2$ . If ${\displaystyle C_{1$ and ${\displaystyle C_{2$ are non-collinear straight line segments, there necessarily exists a point in ${\displaystyle [s_{1},s_{2}]}$ at which ${\displaystyle \mathbf {r} }$ is non-analytic. If the end segments of the smooth interpolation are not straight-segment extensions of line segments ${\displaystyle C_{1$ and ${\displaystyle C_{2$ , ${\displaystyle \mathbf {r} }$ is necessarily non-analytic at ${\displaystyle s_{1$ and ${\displaystyle s_{2$ .

References#edit

Glaister, P. (December 1991), A Flat Function with Some Interesting Properties and an Application, The Mathematical Gazette, Vol. 75, No. 474, pp. 438–440, JSTOR 3618627

This article is mirrored from “Flat function” on English Wikipedia, revision 1381438053, written by Wikipedia's contributors and licensed CC BY-SA 4.0. Editing it here creates a Quire copy with its own history.

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Flat function

Examples of construction of non-trivial flat functions#edit

Construction of univariate flat functions#edit

Construction of multivariate flat functions#edit

A necessary condition for flatness and local non-constancy#edit

Proof#edit

A sufficient condition for flatness#edit

Proof#edit

Flatness of smooth interpolations#edit

See also#edit

References#edit