In
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 ...
,
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 ...
s (also called infinitely
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 ...
functions) and
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 are two very important types of
functions. One can easily prove that any analytic function of a
real
Real may refer to:
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* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010)
...
argument is smooth. The
converse
Converse may refer to:
Mathematics and logic
* Converse (logic), the result of reversing the two parts of a definite or implicational statement
** Converse implication, the converse of a material implication
** Converse nonimplication, a logical c ...
is not true, as demonstrated with the
counterexample
A counterexample is any exception to a generalization. In logic a counterexample disproves the generalization, and does so rigorously in the fields of mathematics and philosophy. For example, the fact that "John Smith is not a lazy student" is a ...
below.
One of the most important applications of smooth functions with
compact support
In mathematics, the support of a real-valued function f is the subset of the function domain containing the elements which are not mapped to zero. If the domain of f is a topological space, then the support of f is instead defined as the smallest ...
is the construction of so-called
mollifier
In mathematics, mollifiers (also known as ''approximations to the identity'') are smooth functions with special properties, used for example in distribution theory to create sequences of smooth functions approximating nonsmooth (generalized) fun ...
s, which are important in theories of
generalized function
In mathematics, generalized functions are objects extending the notion of functions. There is more than one recognized theory, for example the theory of distributions. Generalized functions are especially useful in making discontinuous functions ...
s, such as
Laurent Schwartz
Laurent-Moïse Schwartz (; 5 March 1915 – 4 July 2002) was a French mathematician. He pioneered the theory of distributions, which gives a well-defined meaning to objects such as the Dirac delta function. He was awarded the Fields Medal in 19 ...
's theory of
distribution Distribution may refer to:
Mathematics
*Distribution (mathematics), generalized functions used to formulate solutions of partial differential equations
* Probability distribution, the probability of a particular value or value range of a vari ...
s.
The existence of smooth but non-analytic functions represents one of the main differences between
differential geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...
and
analytic geometry
In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts with synthetic geometry.
Analytic geometry is used in physics and engineerin ...
. In terms of
sheaf theory
In mathematics, a sheaf is a tool for systematically tracking data (such as sets, abelian groups, rings) attached to the open sets of a topological space and defined locally with regard to them. For example, for each open set, the data could ...
, this difference can be stated as follows: the sheaf of differentiable functions on a
differentiable manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
is
fine
Fine may refer to:
Characters
* Sylvia Fine (''The Nanny''), Fran's mother on ''The Nanny''
* Officer Fine, a character in ''Tales from the Crypt'', played by Vincent Spano
Legal terms
* Fine (penalty), money to be paid as punishment for an offe ...
, in contrast with the analytic case.
The functions below are generally used to build up
partitions of unity
In mathematics, a partition of unity of a topological space is a set of continuous functions from to the unit interval ,1such that for every point x\in X:
* there is a neighbourhood of where all but a finite number of the functions of are 0, ...
on differentiable manifolds.
An example function
Definition of the function
Consider the function
:
defined for every
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
''x''.
The function is smooth
The function ''f'' has
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ...
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 of all orders at every point ''x'' of the
real line
In elementary mathematics, a number line is a picture of a graduated straight line (geometry), line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real ...
. The formula for these derivatives is
:
where ''p
n''(''x'') is a
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...
of
degree
Degree may refer to:
As a unit of measurement
* Degree (angle), a unit of angle measurement
** Degree of geographical latitude
** Degree of geographical longitude
* Degree symbol (°), a notation used in science, engineering, and mathematics
...
''n'' − 1 given
recursively
Recursion (adjective: ''recursive'') occurs when a thing is defined in terms of itself or of its type. Recursion is used in a variety of disciplines ranging from linguistics to logic. The most common application of recursion is in mathematics ...
by ''p''
1(''x'') = 1 and
:
for any positive
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
''n''. From this formula, it is not completely clear that the derivatives are continuous at 0; this follows from the
one-sided limit
In calculus, a one-sided limit refers to either one of the two limits of a function f(x) of a real variable x as x approaches a specified point either from the left or from the right.
The limit as x decreases in value approaching a (x approaches ...
:
for any
nonnegative
In mathematics, the sign of a real number is its property of being either positive, negative, or zero. Depending on local conventions, zero may be considered as being neither positive nor negative (having no sign or a unique third sign), or it ...
integer ''m''.
By the
power series representation of the exponential function, we have for every
natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called ''Cardinal n ...
(including zero)
:
because all the positive terms for
are added. Therefore, dividing this inequality by
and taking the
limit from above
In calculus, a one-sided limit refers to either one of the two limits of a function f(x) of a real variable x as x approaches a specified point either from the left or from the right.
The limit as x decreases in value approaching a (x approaches ...
,
:
We now prove the formula for the ''n''th derivative of ''f'' by
mathematical induction
Mathematical induction is a method for proving that a statement ''P''(''n'') is true for every natural number ''n'', that is, that the infinitely many cases ''P''(0), ''P''(1), ''P''(2), ''P''(3), ... all hold. Informal metaphors help ...
. Using the
chain rule
In calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h(x)=f(g(x)) for every , ...
, the
reciprocal rule
In calculus, the reciprocal rule gives the derivative of the reciprocal of a function ''f'' in terms of the derivative of ''f''. The reciprocal rule can be used to show that the power rule holds for negative exponents if it has already been e ...
, and the fact that the derivative of the exponential function is again the exponential function, we see that the formula is correct for the first derivative of ''f'' for all ''x'' > 0 and that ''p''
1(''x'') is a polynomial of degree 0. Of course, the derivative of ''f'' is zero for ''x'' < 0.
It remains to show that the right-hand side derivative of ''f'' at ''x'' = 0 is zero. Using the above limit, we see that
:
The induction step from ''n'' to ''n'' + 1 is similar. For ''x'' > 0 we get for the derivative
:
where ''p''
''n''+1(''x'') is a polynomial of degree ''n'' = (''n'' + 1) − 1. Of course, the (''n'' + 1)st derivative of ''f'' is zero for ''x'' < 0. For the right-hand side derivative of ''f''
(''n'') at ''x'' = 0 we obtain with the above limit
:
The function is not analytic
As seen earlier, the function ''f'' is smooth, and all its derivatives at the
origin
Origin(s) or The Origin may refer to:
Arts, entertainment, and media
Comics and manga
* Origin (comics), ''Origin'' (comics), a Wolverine comic book mini-series published by Marvel Comics in 2002
* The Origin (Buffy comic), ''The Origin'' (Bu ...
are 0. Therefore, the
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 ...
of ''f'' at the origin converges everywhere to the
zero function
0 (zero) is a number representing an empty quantity. In place-value notation such as the Hindu–Arabic numeral system, 0 also serves as a placeholder numerical digit, which works by multiplying digits to the left of 0 by the radix, usuall ...
,
:
and so the Taylor series does not equal ''f''(''x'') for ''x'' > 0. Consequently, ''f'' is not
analytic at the origin.
Smooth transition functions
The function
:
has a strictly positive denominator everywhere on the real line, hence ''g'' is also smooth. Furthermore, ''g''(''x'') = 0 for ''x'' ≤ 0 and ''g''(''x'') = 1 for ''x'' ≥ 1, hence it provides a smooth transition from the level 0 to the level 1 in the
unit interval
In mathematics, the unit interval is the closed interval , that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1. It is often denoted ' (capital letter ). In addition to its role in real analysis, ...
0, 1">/nowiki>0, 1/nowiki>. To have the smooth transition in the real interval ''a'', ''b''">/nowiki>''a'', ''b''/nowiki> with ''a'' < ''b'', consider the function
:
For real numbers , the smooth function
:
equals 1 on the closed interval ''b'', ''c''">/nowiki>''b'', ''c''/nowiki> and vanishes outside the open interval (''a'', ''d''), hence it can serve as a bump function
In mathematics, a bump function (also called a test function) is a function f: \R^n \to \R on a Euclidean space \R^n which is both smooth (in the sense of having continuous derivatives of all orders) and compactly supported. The set of all bump f ...
.
A smooth function which is nowhere real analytic
A more pathological
Pathology is the study of the causal, causes and effects of disease or injury. The word ''pathology'' also refers to the study of disease in general, incorporating a wide range of biology research fields and medical practices. However, when us ...
example is an infinitely differentiable function which is not analytic ''at any point''. It can be constructed by means of a Fourier series
A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or ''p ...
as follows. Define for all
:
Since the series converges for all , this function is easily seen to be of class C∞, by a standard inductive application of the Weierstrass M-test
In mathematics, the Weierstrass M-test is a test for determining whether an infinite series of functions converges uniformly and absolutely. It applies to series whose terms are bounded functions with real or complex values, and is analogous to ...
to demonstrate uniform convergence
In the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence. A sequence of functions (f_n) converges uniformly to a limiting function f on a set E if, given any arbitrarily s ...
of each series of derivatives.
We now show that is not analytic at any dyadic rational
In mathematics, a dyadic rational or binary rational is a number that can be expressed as a fraction whose denominator is a power of two. For example, 1/2, 3/2, and 3/8 are dyadic rationals, but 1/3 is not. These numbers are important in compute ...
multiple of π, that is, at any with and . Since the sum of the first terms is analytic, we need only consider , the sum of the terms with . For all orders of derivation with , and we have
:
where we used the fact that for all , and we bounded the first sum from below by the term with . As a consequence, at any such
:
so that the radius of convergence
In mathematics, the radius of convergence of a power series is the radius of the largest disk at the center of the series in which the series converges. It is either a non-negative real number or \infty. When it is positive, the power series co ...
of the 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 ...
of at is 0 by the Cauchy-Hadamard formula. Since the set of analyticity of a function is an open set, and since dyadic rationals are dense
Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematically ...
, we conclude that , and hence , is nowhere analytic in .
Application to Taylor series
For every sequence α0, α1, α2, . . . of real or complex numbers, the following construction shows the existence of a smooth function ''F'' on the real line which has these numbers as derivatives at the origin. In particular, every sequence of numbers can appear as the coefficients of the 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 ...
of a smooth function. This result is known as Borel's lemma
In mathematics, Borel's lemma, named after Émile Borel, is an important result used in the theory of asymptotic expansions and partial differential equations.
Statement
Suppose ''U'' is an open set in the Euclidean space R''n'', and suppose that ...
, after Émile Borel
Félix Édouard Justin Émile Borel (; 7 January 1871 – 3 February 1956) was a French mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems.
Math ...
.
With the smooth transition function ''g'' as above, define
:
This function ''h'' is also smooth; it equals 1 on the closed interval −1,1">/nowiki>−1,1/nowiki> and vanishes outside the open interval (−2,2). Using ''h'', define for every natural number ''n'' (including zero) the smooth function
:
which agrees with the monomial
In mathematics, a monomial is, roughly speaking, a polynomial which has only one term. Two definitions of a monomial may be encountered:
# A monomial, also called power product, is a product of powers of variables with nonnegative integer exponent ...
''xn'' on −1,1">/nowiki>−1,1/nowiki> and vanishes outside the interval (−2,2). Hence, the ''k''-th derivative of ''ψn'' at the origin satisfies
:
and the boundedness theorem
In calculus, the extreme value theorem states that if a real-valued function f is continuous on the closed interval ,b/math>, then f must attain a maximum and a minimum, each at least once. That is, there exist numbers c and d in ,b/math> suc ...
implies that ''ψn'' and every derivative of ''ψn'' is bounded. Therefore, the constants
:
involving the supremum norm
In mathematical analysis, the uniform norm (or ) assigns to real- or complex-valued bounded functions defined on a set the non-negative number
:\, f\, _\infty = \, f\, _ = \sup\left\.
This norm is also called the , the , the , or, when the ...
of ''ψn'' and its first ''n'' derivatives, are well-defined real numbers. Define the scaled functions
:
By repeated application of the chain rule
In calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h(x)=f(g(x)) for every , ...
,
:
and, using the previous result for the ''k''-th derivative of ''ψn'' at zero,
:
It remains to show that the function
:
is well defined and can be differentiated term-by-term infinitely many times.[See e.g. Chapter V, Section 2, Theorem 2.8 and Corollary 2.9 about the differentiability of the limits of sequences of functions in ] To this end, observe that for every ''k''
:
where the remaining infinite series converges by the ratio test
In mathematics, the ratio test is a test (or "criterion") for the convergence of a series
:\sum_^\infty a_n,
where each term is a real or complex number and is nonzero when is large. The test was first published by Jean le Rond d'Alembert and ...
.
Application to higher dimensions
For every radius ''r'' > 0,
:
with Euclidean norm
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean s ...
, , ''x'', , defines a smooth function on ''n''-dimensional Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
with support
Support may refer to:
Arts, entertainment, and media
* Supporting character
Business and finance
* Support (technical analysis)
* Child support
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Construction
* Support (structure), or lateral support, a ...
in the ball
A ball is a round object (usually spherical, but can sometimes be ovoid) with several uses. It is used in ball games, where the play of the game follows the state of the ball as it is hit, kicked or thrown by players. Balls can also be used f ...
of radius ''r'', but .
Complex analysis
This pathology cannot occur with differentiable functions of a complex variable
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebrai ...
rather than of a real variable. Indeed, all holomorphic functions are analytic
In complex analysis, a complex-valued function f of a complex variable z:
*is said to be holomorphic at a point a if it is differentiable at every point within some open disk centered at a, and
* is said to be analytic at a if in some open disk ...
, so that the failure of the function ''f'' defined in this article to be analytic in spite of its being infinitely differentiable is an indication of one of the most dramatic differences between real-variable and complex-variable analysis.
Note that although the function ''f'' has derivatives of all orders over the real line, the analytic continuation
In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a new ...
of ''f'' from the positive half-line ''x'' > 0 to the complex plane
In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
, that is, the function
:
has an essential singularity
In complex analysis, an essential singularity of a function is a "severe" singularity near which the function exhibits odd behavior.
The category ''essential singularity'' is a "left-over" or default group of isolated singularities that a ...
at the origin, and hence is not even continuous, much less analytic. By the great Picard theorem, it attains every complex value (with the exception of zero) infinitely many times in every neighbourhood of the origin.
See also
* Bump function
In mathematics, a bump function (also called a test function) is a function f: \R^n \to \R on a Euclidean space \R^n which is both smooth (in the sense of having continuous derivatives of all orders) and compactly supported. The set of all bump f ...
* Fabius function
* Flat function
In mathematics, especially real analysis, a flat function is a smooth function f : \mathbb \to \mathbb all of whose derivatives vanish (mathematics), vanish at a given point x_0 \in \mathbb. The flat functions are, in some sense, the antitheses of ...
* Mollifier
In mathematics, mollifiers (also known as ''approximations to the identity'') are smooth functions with special properties, used for example in distribution theory to create sequences of smooth functions approximating nonsmooth (generalized) fun ...
Notes
External links
* {{planetmath reference, urlname=InfinitelydifferentiableFunctionThatIsNotAnalytic, title=Infinitely-differentiable function that is not analytic
Smooth functions
Articles containing proofs