mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, especially

real analysis In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include converg ...

, a flat function is a

smooth function In mathematical analysis, the smoothness of a function (mathematics), function is a property measured by the number of Continuous function, continuous Derivative (mathematics), derivatives it has over some domain, called ''differentiability cl ...

f : \mathbb \to \mathbb

all of whose

derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. F ...

s vanish at a given point

x_0 \in \mathbb

. The flat functions are, in some sense, the

antitheses Antithesis (Greek for "setting opposite", from "against" and "placing") is used in writing or speech either as a proposition that contrasts with or reverses some previously mentioned proposition, or when two opposites are introduced together f ...

of the

analytic function In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex an ...

s. An analytic function

f : \mathbb \to \mathbb

is given by a convergent

power series In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots where ''an'' represents the coefficient of the ''n''th term and ''c'' is a const ...

close to some point

x_0 \in \mathbb

: :

f(x) \sim \lim_ \sum_^n \frac (x-x_0)^k .

In the case of a flat function, all derivatives vanish at

x_0 \in \mathbb

, i.e.

f^(x_0) = 0

for all

k \in \mathbb

. This means that a meaningful

Taylor series In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor serie ...

expansion in a

neighbourhood A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural are ...

x_0

is impossible. In the language of

Taylor's theorem In calculus, Taylor's theorem gives an approximation of a ''k''-times differentiable function around a given point by a polynomial of degree ''k'', called the ''k''th-order Taylor polynomial. For a smooth function, the Taylor polynomial is the t ...

, the non-constant part of the function always lies in the remainder

R_n(x)

for all

n \in \mathbb

. The function need not be flat at just one point. Trivially,

constant function In mathematics, a constant function is a function whose (output) value is the same for every input value. For example, the function is a constant function because the value of is 4 regardless of the input value (see image). Basic properties ...

s on

\mathbb

are flat everywhere. But there are also other, less trivial, examples.

Example

The function defined by :

f(x) = \begin
e^ & \textx\neq 0 \\
0 & \textx = 0
\end

is flat at

x = 0

. Thus, this is an example of a

non-analytic smooth function In mathematics, smooth functions (also called infinitely differentiable functions) and analytic functions are two very important types of functions. One can easily prove that any analytic function of a real argument is smooth. The converse is not ...

. The pathological nature of this example is partially illuminated by the fact that its extension to the

complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...

s is, in fact, not

differentiable In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its ...

References

* {{Citation, first=P., last=Glaister, title=A Flat Function with Some Interesting Properties and an Application, publisher=The Mathematical Gazette, Vol. 75, No. 474, pp. 438–440 , date=December 1991, jstor=3618627 Real analysis Algebraic geometry Differential calculus Smooth functions Differential structures