Flat function

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.

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.

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 ${\displaystyle g(x)=e^{-|x-x_{0}|^{-a}}}$

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

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 ${\displaystyle \mathbf {x} _{0}}$.

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 ${\displaystyle x_{0}}$.

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 ${\displaystyle x_{0}}$.

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 ${\displaystyle x_{0}}$.

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.

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 ${\displaystyle s_{2}}$.