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Difference between revisions of "Smooth function"

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A function for which each value of the argument is a smooth point (cf. [[Smooth point of a function|Smooth point of a function]]). A smooth function can be discontinuous. If a smooth function is continuous on an interval, the set of its points of differentiability is dense in the interval and has the cardinality of the continuum. There exist continuous smooth functions on the real axis that are not almost-everywhere differentiable. A smooth function has a derivative at each point of local extremum; as a result, the basic theorems of differential calculus, the theorems of Rolle, Lagrange, Cauchy, Darboux, etc., remain valid for smooth continuous functions.
 
A function for which each value of the argument is a smooth point (cf. [[Smooth point of a function|Smooth point of a function]]). A smooth function can be discontinuous. If a smooth function is continuous on an interval, the set of its points of differentiability is dense in the interval and has the cardinality of the continuum. There exist continuous smooth functions on the real axis that are not almost-everywhere differentiable. A smooth function has a derivative at each point of local extremum; as a result, the basic theorems of differential calculus, the theorems of Rolle, Lagrange, Cauchy, Darboux, etc., remain valid for smooth continuous functions.
  
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Notice that any additive function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085870/s0858701.png" /> (i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085870/s0858702.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085870/s0858703.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085870/s0858704.png" />) is smooth. There exist additive functions that are continuous at no point.
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Notice that any additive function $f$ (i.e. $f(x+y)=f(x)+f(y)$ for all $x$ and $y$) is smooth. There exist additive functions that are continuous at no point.
  
The notion of a smooth function as introduced above is a rather uncommon one. Usually  "smooth function"  means  "sufficient often differentiable function" , most often <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085870/s0858705.png" />-function (infinitely often differentiable function); it can also mean  "having modulus of smoothness satisfying certain growth conditions"  (cf. also [[Smoothness, modulus of|Smoothness, modulus of]]).
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The notion of a smooth function as introduced above is a rather uncommon one. Usually  "smooth function"  means  "sufficient often differentiable function", most often $C^\infty$-function (infinitely often differentiable function); it can also mean  "having modulus of smoothness satisfying certain growth conditions"  (cf. also [[Smoothness, modulus of|Smoothness, modulus of]]).

Latest revision as of 06:02, 22 August 2014

A function for which each value of the argument is a smooth point (cf. Smooth point of a function). A smooth function can be discontinuous. If a smooth function is continuous on an interval, the set of its points of differentiability is dense in the interval and has the cardinality of the continuum. There exist continuous smooth functions on the real axis that are not almost-everywhere differentiable. A smooth function has a derivative at each point of local extremum; as a result, the basic theorems of differential calculus, the theorems of Rolle, Lagrange, Cauchy, Darboux, etc., remain valid for smooth continuous functions.


Comments

Notice that any additive function $f$ (i.e. $f(x+y)=f(x)+f(y)$ for all $x$ and $y$) is smooth. There exist additive functions that are continuous at no point.

The notion of a smooth function as introduced above is a rather uncommon one. Usually "smooth function" means "sufficient often differentiable function", most often $C^\infty$-function (infinitely often differentiable function); it can also mean "having modulus of smoothness satisfying certain growth conditions" (cf. also Smoothness, modulus of).

How to Cite This Entry:
Smooth function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Smooth_function&oldid=12211
This article was adapted from an original article by V.F. Emel'yanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article