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\neq0 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\neq0. Since F is infinitely differentiable at x_{0}, then F_{k} is continuous at x_{0}. Since r\neq0, 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\neq0, 0\notin(\operatorname{min}\{r/2,3r/2\},\operatorname{max}\{r/2,3r/2\}), so F_{k}(x)\neq0, 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

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