Flat function

Short description: Function whose all derivatives vanish at a point

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

In real analysis, a real function is flat at a point $x_0$ in the interior of its domain if all its derivatives at $x_0$ exist and equal $0.$

A real function is constant in a neighbourhood of a point $x_0$ in the interior of its domain if and only if the function is flat at $x_0$ and is analytic at $x_0$.

An example of a function is flat at only at an isolated point is $f:\mathbb{ℝ}\to \mathbb{ℝ}$ such that $f(0)=0$ and that for all $x\in \mathbb{ℝ}$, $x\ne 0$ 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 $\mathbb{ℂ}$, that is the function $f_{ext}:\mathbb{ℂ}\to \mathbb{ℂ}$ such that $f_{ext}(0)=0$ and that for all $z\in \mathbb{ℂ}$, $z\ne 0$ implies $f_{ext}(z)=\exp (-z^{-2})$, is not holomorphic at $0$, due to that for complex functions, holomorphicity at a point implies analyticity at that point.

Flatness of bump functions

A bump function is a function, with domain ${\mathbb{ℝ}}^n$ and codomain $\mathbb{ℝ}$, such that it is smooth (infinitely continuously differentiable) on ${\mathbb{ℝ}}^n$, and has bounded support, that is, the set of points in ${\mathbb{ℝ}}^n$ that are mapped to a non-zero value is a bounded set.

A bump function is flat and non-analytic at each boundary point of the closure of its support.

Let $\mathbf{𝐛}$ be a boundary point of the closure of the support of a bump function $F:{\mathbb{ℝ}}^n\to \mathbb{ℝ}$.

Proof of flatness of $F$ at $\mathbf{𝐛}$

If there existed any $k\in \mathbb{\mathbb{N}}$ such that a $k$-th partial derivative of $F$ (call it $F_k$) at $\mathbf{𝐛}$ is a non-zero real number, say $r$, there would need to exist a positive real number $\delta$ such that for all $\mathbf{𝐱}\in {\mathbb{ℝ}}^n$ such that $||\mathbf{𝐱}-\mathbf{𝐛}||<\delta$, $|F_k(\mathbf{𝐱})-F_k(\mathbf{𝐛})|<|r|/2$, or in other words, $F_k(\mathbf{𝐱})$ is between $r/2$ and $3r/2$; this is an implication of the continuity of $F_k$ at $\mathbf{𝐛}$.

This necessitates the existence of a neighbourhood of $\mathbf{𝐛}$ that is a subset of the support of $F_k$, and hence also a subset of the closure of the support of $F$, since everywhere outside the closure of the support of $F$, $F_k$ evaluates to $0$.

This contradicts that $\mathbf{𝐛}$ is a boundary point of the closure of the support of $F$.

Hence, there does not exist any $k\in \mathbb{\mathbb{N}}$ such that $F_k(\mathbf{𝐛})$ is a non-zero real number. In other words, $F$ is flat at $\mathbf{𝐛}$.

Proof of non-analyticity of $F$ at $\mathbf{𝐛}$

Since $F$ is flat at $\mathbf{𝐛}$ (as shown above), the Taylor series of $F$ at $\mathbf{𝐛}$ is zero in a neighbourhood of $\mathbf{𝐛}$.

If $F$ is analytic at $\mathbf{𝐛}$, then there exists a neighbourhood $N$ of $\mathbf{𝐛}$ such that for all $\mathbf{𝐱}\in N$, $F(\mathbf{𝐱})=0$.

Since $\mathbf{𝐛}$ is a boundary point of the closure of the support of $F$, every neighbourhood of $\mathbf{𝐛}$ must contain at least one point $\mathbf{𝐱}$ such that $F(\mathbf{𝐱})\ne 0$. This contradicts the existence of a neighbourhood $N$ of $\mathbf{𝐛}$ such that for all $\mathbf{𝐱}\in N$, $F(\mathbf{𝐱})=0$.

Hence, $F$ is non-analytic at $\mathbf{𝐛}$.

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