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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 $0$ .

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

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

Examples of construction of non-trivial flat functions

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

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

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

Construction of multivariate flat functions

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

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

A necessary condition for flatness and local non-constancy

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

Proof

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

A sufficient condition for flatness

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

Proof

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

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

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

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

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

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

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

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

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

References

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

Sample Page

Examples of construction of non-trivial flat functions

Construction of univariate flat functions

Construction of multivariate flat functions

A necessary condition for flatness and local non-constancy

Proof

A sufficient condition for flatness

Proof

Flatness of smooth interpolations

See also

References