Flat function

The graph of $f:ℝ\to ℝ$ such that $f(0)=0$ and that for all $x\in ℝ$, $x\ne 0$ 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 $f:ℝ\to ℝ$ such that $f(0)=0$ and that for all $x\in ℝ$, $x\ne 0$ 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 $ℂ$ 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 $g:S\to ℝ$ (where $S\subseteq ℝ$ is a neighbourhood of a point $x_0\in ℝ$) be such that $g(x_0)=0$ and that for all $x\in S$, $x\ne x_0$ implies $g(x)=\exp (-|x-x_0{|}^{-a}).$

Then $g$ is flat at $x_0$.

Construction of multivariate flat functions

Let $G:ℝ\to ℝ$ be flat at $0$, and let $H:P\to ℝ$ (where $n\in \mathbb{N}$, ${𝐱}_0$ is an $n$-dimensional real coordinate vector, and $P\subseteq {ℝ}^n$ is a neighbourhood of ${𝐱}_0$) be such that for all $𝐱\in P$,$H(𝐱)=G(||𝐱-{𝐱}_0||)$, where for all $𝐩\in {ℝ}^n$, $||𝐩||$ denotes the Euclidean norm of $𝐩$.

Then $H$ is flat at ${𝐱}_0$.

A necessary condition for flatness and local non-constancy

Let $S\subseteq {ℝ}^n$ for some $n\in \mathbb{N}$ and let $F:S\to ℝ$ 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 $F(x)\ne F(x_0)$, 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 $F(x)\ne F(x_0)$. Hence, by contradiction, $F$ is non-analytic at $x_0$.

A sufficient condition for flatness

Let $S\subseteq {ℝ}^n$ for some $n\in \mathbb{N}$ and let $F:S\to ℝ$ 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 $r\in ℝ$ such that $r\ne 0$. Since $F$ is infinitely differentiable at $x_0$, then $F_k$ is continuous at $x_0$. Since $r\ne 0$, 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 $(\min \{r/2,3r/2\},\max \{r/2,3r/2\})$. Since $r\ne 0$, $0\notin (\min \{r/2,3r/2\},\max \{r/2,3r/2\})$, so $F_k(x)\ne 0$, 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 $s_1\in ℝ$ and $s_2\in ℝ$ be such that $s_1<s_2$.

Let $I_1\subset ℝ$ be an interval with non-empty interior, with supremum $s_1$, and containing $s_1$; and let $I_2\subset ℝ$ 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 ${𝐫}_1:I_1\to {ℝ}^n$ 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 $||{𝐫}_{1^{\prime }}(s)||=1$ for all $s\in \mathrm{int}(I_1)$. Let ${𝐫}_2:I_2\to {ℝ}^n$ 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 $||{𝐫}_{2^{\prime }}(s)||=1$ for all $s\in \mathrm{int}(I_2)$.

Let curves $C_1$ and $C_2$ be the images of the domains of ${𝐫}_1$ and ${𝐫}_2$, respectively. Both $C_1$ and $C_2$ inhabit ${ℝ}^n$.

A smooth interpolation between $C_1$ and $C_2$, between the points ${𝐫}_1(s_1)$ and ${𝐫}_2(s_2)$, is the image of the domain of a function ${𝐫}_0:(s_1,s_2)\to {ℝ}^n$ such that the left-hand limit of ${𝐫}_0$ at $s_1$ is ${𝐫}_1(s_1)$, the right-hand limit of ${𝐫}_0$ at $s_2$ is ${𝐫}_2(s_2)$, and for all $k\in \mathbb{N}$, the left-hand limit of the $k$-th derivative of ${𝐫}_0$ at $s_1$ is equal to the right-hand limit of the $k$-th derivative of ${𝐫}_1$ at $s_1$, and the right-hand limit of the $k$-th derivative of ${𝐫}_0$ at $s_2$ is equal to the left-hand limit of the $k$-th derivative of ${𝐫}_2$ at $s_2$. A smooth interpolation between $C_1$ and $C_2$ is defined to have $G^{\mathrm{\infty }}$ continuity (geometric continuity of all orders) with $C_1$ and $C_2$.

Let $𝐫:I_1\cup (s_1,s_2)\cup I_2\to {ℝ}^n$ be such that: for all $s\in I_1$, $𝐫(s)={𝐫}_1(s)$; for all $s\in (s_1,s_2)$, $𝐫(s)={𝐫}_0(s)$; and for all $s\in I_2$, $𝐫(s)={𝐫}_2(s)$.

If $C_1$ and $C_2$ are straight line segments, $𝐫$ 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 $𝐫$ is non-analytic. If the end segments of the smooth interpolation are not straight-segment extensions of line segments $C_1$ and $C_2$, $𝐫$ is necessarily non-analytic at $s_1$ and $s_2$.

See also

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

References

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