Flat function

The graph of $f:\mathbb {R} \to \mathbb {R}$ such that $f(0)=0$ and that for all $x\in \mathbb {R}$, $x\neq 0$ implies $f(x)=e^{-1/x^{2}}$

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

See also: Smoothness § Geometric continuity

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 $s_{2}$.

See also

• Bump function
• Continuous function
• Differentiable function
• Smoothness
• Analytic function
• Support (mathematics)

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