Flat function

In real analysis, a real function is defined to be flat at a point in the interior of its domain if and only if all its derivatives or partial derivatives exist at that point and equal 0 {\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 f : R → R {\displaystyle f:\mathbb {R} \to \mathbb {R} } such that f ( 0 ) = 0 {\displaystyle f(0)=0} and that for all x ∈ R {\displaystyle x\in \mathbb {R} }, x ≠ 0 {\displaystyle x\neq 0} implies f ( x ) = e − 1 / x 2 {\displaystyle f(x)=e^{-1/x^{2}}}; the function f {\displaystyle f} is flat only at 0 {\displaystyle 0}.

Since f {\displaystyle f} is not analytic at 0 {\displaystyle 0}, the extension of f {\displaystyle f} to C {\displaystyle \mathbb {C} } is not holomorphic at 0 {\displaystyle 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 {\displaystyle a} be a positive real number and let g : S → R {\displaystyle g:S\to \mathbb {R} } (where S ⊆ R {\displaystyle S\subseteq \mathbb {R} } is a neighbourhood of a point x 0 ∈ R {\displaystyle x_{0}\in \mathbb {R} }) be such that g ( x 0 ) = 0 {\displaystyle g(x_{0})=0} and that for all x ∈ S {\displaystyle x\in S}, x ≠ x 0 {\displaystyle x\neq x_{0}} implies g ( x ) = e − | x − x 0 | − a {\displaystyle g(x)=e^{-|x-x_{0}|^{-a}}}

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

Construction of multivariate flat functions

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

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

Flatness of bump functions

A bump function is a function, with domain R n {\displaystyle \mathbb {R} ^{n}} and codomain R {\displaystyle \mathbb {R} }, such that it is smooth (infinitely continuously differentiable) on R n {\displaystyle \mathbb {R} ^{n}}, and has bounded support, that is, the set of points in R n {\displaystyle \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 b {\displaystyle \mathbf {b} } be a boundary point of the closure of the support of a bump function F : R n → R {\displaystyle F:\mathbb {R} ^{n}\to \mathbb {R} }.

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

Assume the existence of a k ∈ N {\displaystyle k\in \mathbb {N} } such that a k {\displaystyle k}-th partial derivative of F {\displaystyle F} (call it F k {\displaystyle F_{k}}) at b {\displaystyle \mathbf {b} } is a non-zero real number, say r {\displaystyle r}. Since F {\displaystyle F} is infinitely continuously differentiable at b {\displaystyle \mathbf {b} }, then F k {\displaystyle F_{k}} is continuous at b {\displaystyle \mathbf {b} }. Since r ≠ 0 {\displaystyle r\neq 0}, | r | / 2 > 0 {\displaystyle |r|/2>0}. Then there would need to exist a positive real number δ {\displaystyle \delta } such that for all x ∈ R n {\displaystyle \mathbf {x} \in \mathbb {R} ^{n}} such that | | x − b | | < δ {\displaystyle ||\mathbf {x} -\mathbf {b} ||<\delta }, | F k ( x ) − F k ( b ) | < | r | / 2 {\displaystyle |F_{k}(\mathbf {x} )-F_{k}(\mathbf {b} )|<|r|/2}, or in other words, F k ( x ) {\displaystyle F_{k}(\mathbf {x} )} lies in the open interval ( min ⁡ { r / 2 , 3 r / 2 } , max ⁡ { r / 2 , 3 r / 2 } ) . {\displaystyle (\operatorname {min} \{r/2,3r/2\},\operatorname {max} \{r/2,3r/2\}).}

Since the open interval ( min ⁡ { r / 2 , 3 r / 2 } , max ⁡ { r / 2 , 3 r / 2 } ) {\displaystyle (\operatorname {min} \{r/2,3r/2\},\operatorname {max} \{r/2,3r/2\})} is non-empty and does not contain 0 {\displaystyle 0} (whether or not r {\displaystyle r} is positive or negative, as long as r ≠ 0 {\displaystyle r\neq 0}), this necessitates the existence of a neighbourhood of b {\displaystyle \mathbf {b} } that is a subset of the support of F k {\displaystyle F_{k}}, and hence also a subset of the closure of the support of F {\displaystyle F}, since everywhere outside the closure of the support of F {\displaystyle F}, F {\displaystyle F} evaluates to 0 {\displaystyle 0} and hence F k {\displaystyle F_{k}} evaluates to 0 {\displaystyle 0}.

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

Hence, there does not exist any k ∈ N {\displaystyle k\in \mathbb {N} } such that F k ( b ) {\displaystyle F_{k}(\mathbf {b} )} is a non-zero real number. In other words, for all k ∈ N {\displaystyle k\in \mathbb {N} }, F k ( b ) = 0 {\displaystyle F_{k}(\mathbf {b} )=0}.

Hence, F {\displaystyle F} is flat at b {\displaystyle \mathbf {b} }.

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

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

Assume that F {\displaystyle F} is analytic at b {\displaystyle \mathbf {b} }. Then there exists a neighbourhood N {\displaystyle N} of b {\displaystyle \mathbf {b} } such that for all x ∈ N {\displaystyle \mathbf {x} \in N}, F ( x ) = 0 {\displaystyle F(\mathbf {x} )=0}.

Since b {\displaystyle \mathbf {b} } is a boundary point of the closure of the support of F {\displaystyle F}, so b {\displaystyle \mathbf {b} } is a boundary point of the support of F {\displaystyle F} (since boundary of the closure of a set is a subset of the boundary of the set). Hence, every neighbourhood of b {\displaystyle \mathbf {b} } must contain at least one point x {\displaystyle \mathbf {x} } such that F ( x ) ≠ 0 {\displaystyle F(\mathbf {x} )\neq 0}. This contradicts the existence of a neighbourhood N {\displaystyle N} of b {\displaystyle \mathbf {b} } such that for all x ∈ N {\displaystyle \mathbf {x} \in N}, F ( x ) = 0 {\displaystyle F(\mathbf {x} )=0}.

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

Flatness of smooth interpolations

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

Let I 1 ⊂ R {\displaystyle I_{1}\subset \mathbb {R} } be an interval with non-empty interior, with supremum s 1 {\displaystyle s_{1}}, and containing s 1 {\displaystyle s_{1}}; and let I 2 ⊂ R {\displaystyle I_{2}\subset \mathbb {R} } be an interval with non-empty interior, with infimum s 2 {\displaystyle s_{2}}, and containing s 2 {\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 n ∈ N {\displaystyle n\in \mathbb {N} }. Let r 1 : I 1 → R n {\displaystyle \mathbf {r} _{1}:I_{1}\to \mathbb {R} ^{n}} be continuously differentiable at every point in the interior of I 1 {\displaystyle I_{1}}, left-continuous at s 1 {\displaystyle s_{1}} and have the left-hand limit of its derivatives of all orders be finite at s 1 {\displaystyle s_{1}}; also let | | r 1 ′ ( s ) | | = 1 {\displaystyle ||\mathbf {r} _{1}'(s)||=1} for all s ∈ int ⁡ ( I 1 ) {\displaystyle s\in \operatorname {int} (I_{1})}. Let r 2 : I 2 → R n {\displaystyle \mathbf {r} _{2}:I_{2}\to \mathbb {R} ^{n}} be continuously differentiable at every point in the interior of I 2 {\displaystyle I_{2}}, right-continuous at s 2 {\displaystyle s_{2}} and have the right-hand limit of its derivatives of all orders be finite at s 2 {\displaystyle s_{2}}; also let | | r 2 ′ ( s ) | | = 1 {\displaystyle ||\mathbf {r} _{2}'(s)||=1} for all s ∈ int ⁡ ( I 2 ) {\displaystyle s\in \operatorname {int} (I_{2})}.

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

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

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

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