Smooth function definition and examples. Smooth function

The points of its differentiation are dense on it and have a continuum. There are continuous functions that are smooth on the number line and are not differentiable. G. f. has a derivative at every point local extremum and, due to this, for smooth continuous functions the main theorems remain valid differential calculus- the theorems of Rolle, Lagrange, Cauchy,

Darboux and others. V. F. Emelyanov.

Mathematical encyclopedia. - M.: Soviet Encyclopedia. I. M. Vinogradov. 1977-1985.

See what "SMOOTH FUNCTION" is in other dictionaries:

Or a continuously differentiable function is a function that has a continuous derivative over the entire definition set. Basic information Smooth functions of higher orders are also considered, namely, a function with an order of smoothness has ... ... Wikipedia

Smooth function- a function all partial derivatives of which, up to order r inclusive, are continuous. This means "smoothness of order r." ...

smooth function- A function whose partial derivatives, up to order r inclusive, are continuous. This means "smoothness of order r". Topics: economics EN smooth function… Technical Translator's Guide

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1) P. f. in the theory of trigonometric series, a function introduced by B. Riemann (B. Riemann, 1851) (see) to study the issue on the representability of a trigonometric function. near. Let a series (*) be given with limited... ... Mathematical Encyclopedia

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On the entire set of definitions. Very often under smooth functions mean functions that have continuous derivatives of all orders.

Basics[ | ]

Smooth functions of higher orders are also considered, namely, a function with order of smoothness r ⩾ 0 (\displaystyle r\geqslant 0) has continuous derivatives of all orders up to r (\displaystyle r) inclusive (the zeroth order derivative is the function itself). Such functions are called r (\displaystyle r)-smooth. Many r (\displaystyle r)-smooth functions defined in the domain are denoted by C r (Ω) (\displaystyle C^(r)(\Omega)). Record f ∈ C ∞ (Ω) (\displaystyle f\in C^(\infty )(\Omega)) means that f ∈ C r (Ω) (\displaystyle f\in C^(r)(\Omega)) for anyone r (\displaystyle r), such functions are called endlessly-smooth(sometimes under smooth functions they mean infinitely smooth). Sometimes the notation is also used f ∈ C ω (Ω) (\displaystyle f\in C^(\omega )(\Omega)) or f ∈ C a (Ω) (\displaystyle f\in C^(a)(\Omega)), which means that f (\displaystyle f)- analytical.

For example, C 0 (Ω) (\displaystyle C^(0)(\Omega))- a set of continuous Ω (\displaystyle \Omega ) functions, and C 1 (Ω) (\displaystyle C^(1)(\Omega))- a set of continuously differentiable on Ω (\displaystyle \Omega ) functions, that is, functions that have a continuous derivative at every point in this region.

If the order of smoothness is not specified, then it is usually assumed to be sufficient for all actions performed on the function in the course of the current reasoning to make sense.

Approximation by analytical functions[ | ]

Let the area Ω (\displaystyle \Omega ) open at R n (\displaystyle \mathbb (R) ^(n)) And f ∈ C k (Ω) (\displaystyle f\in C^(k)(\Omega)), 0 ⩽ k ⩽ ∞ (\displaystyle 0\leqslant k\leqslant \infty ). Let ( K p ) (\displaystyle $K_(p)$)- sequence of compact subsets Ω (\displaystyle \Omega ) such that K 0 = ∅ (\displaystyle K_(0)=\varnothing ), K p ⊂ K p + 1 (\displaystyle K_(p)\subset K_(p+1)) And ⋃ K p = Ω (\displaystyle \bigcup K_(p)=\Omega ). Let ( n p ) (\displaystyle $n_(p)$)- an arbitrary sequence of positive integers and m p = min (k , n p) (\displaystyle m_(p)=\min(k,\;n_(p))). Finally, let ( ε p ) (\displaystyle $\varepsilon _(p)$)- arbitrary sequence positive numbers. Then there is a real-analytic function g (\displaystyle g), defined in Ω (\displaystyle \Omega ) such that for everyone p ⩾ 0 (\displaystyle p\geqslant 0) the inequality is satisfied

‖ f − g ‖ C m p (K p + 1 ∖ K p)< ε p , {\displaystyle \|f-g\|_{C^{m_{p}}({K_{p+1}\backslash K_{p}})}<\varepsilon _{p},}

Where ‖ f − g ‖ C m p (K p + 1 ∖ K p) (\displaystyle \|f-g\|_(C^(m_(p))((K_(p+1)\backslash K_(p))) )) denotes the maximum of the norms (in the sense of uniform convergence, that is, the maximum of the modulus on the set K p + 1 ∖ K p (\displaystyle (K_(p+1)\backslash K_(p)))) derivatives of the function f − g (\displaystyle f-g) all orders from zero to m p (\displaystyle (m_(p))) inclusive.

On the entire set of definitions.

Basics

Smooth functions of higher orders are also considered, namely, a function with order of smoothness $r$ has a continuous derivative of order $r$ . Many such functions defined in the field $\Omega$ denoted by $C^r(\Omega)$ . $f\in C^\infty(\Omega)$ means that $f\in C^r(\Omega)$ for anyone $r$ , A $f\in C^\omega(\Omega)=C^a(\Omega)$ means that $f$ - analytical.

For example, $C^0(\Omega)$ - a set of continuous $\Omega$ functions, and $C^1(\Omega)$ - a set of continuously differentiable on $\Omega$ functions, i.e. functions having a continuous derivative at each point of this region.

If the order of smoothness is not specified, then it is usually assumed to be sufficient for all actions performed on the function in the course of the current reasoning to make sense.

For fine class analysis differentiable functions also introduce the concept fractional smoothness at a point or Hölder exponent, which generalizes all the above concepts of smoothness.

Function $f$ belongs to class $C^(r,\;\alpha)$ , Where $r$ is a non-negative integer and $0<\alpha\leqslant 1$ , if it has derivatives up to order $r$ inclusive and $f^((r))$ is Hölder with exponent $\alpha$ .

In translated literature, along with the term "Hölder exponent", the term “Lipschitz exponent” is used.

Approximation of continuously differentiable functions by analytic ones

Let $\Omega$ open in $\R^n$ And $f\in C^k(\Omega)$ , $0\leqslant k\leqslant\infty$ . Let $$K_p$$ - sequence of compact subsets $\Omega$ such that $K_0=\varnothing$ , $K_p\subset K_(p+1)$ And $\bigcup K_p=\Omega$ . Let $$n_p$$ - an arbitrary sequence of positive integers and $m_p=\min(k,\;n_p)$ . Finally, let $$\varepsilon_p$$ - an arbitrary sequence of positive numbers. Then there is $\R$ -analytical function $g$ V $\Omega$ such that for everyone $p\geqslant 0$ :

$||f-g||^(K_(p+1)\backslash K_p)_(m_p)<\varepsilon_p.$

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An excerpt characterizing the smooth function

When Nikolushka was taken away, Princess Marya went up to her brother again, kissed him and, unable to resist any longer, began to cry.
He looked at her intently.
-Are you talking about Nikolushka? - he said.
Princess Marya, crying, bowed her head affirmatively.
“Marie, you know Evan...” but he suddenly fell silent.
-What are you saying?
- Nothing. There’s no need to cry here,” he said, looking at her with the same cold gaze.

When Princess Marya began to cry, he realized that she was crying that Nikolushka would be left without a father. With great effort he tried to return to life and was transported to their point of view.
“Yes, they must find it pathetic! - he thought. “How simple it is!”
“The birds of the air neither sow nor reap, but your father feeds them,” he said to himself and wanted to say the same to the princess. “But no, they will understand it in their own way, they will not understand! What they cannot understand is that all these feelings that they value are all ours, all these thoughts that seem so important to us are that they are not needed. We can't understand each other." - And he fell silent.

Prince Andrei's little son was seven years old. He could barely read, he didn't know anything. He experienced a lot after this day, acquiring knowledge, observation, and experience; but if he had then possessed all these later acquired abilities, he could not have understood better, more deeply the full meaning of that scene that he saw between his father, Princess Marya and Natasha than he understood it now. He understood everything and, without crying, left the room, silently approached Natasha, who followed him out, and shyly looked at her with thoughtful, beautiful eyes; his raised, rosy upper lip trembled, he leaned his head against it and began to cry.
From that day on, he avoided Desalles, avoided the countess who caressed him, and either sat alone or timidly approached Princess Marya and Natasha, whom he seemed to love even more than his aunt, and quietly and shyly caressed them.
Princess Marya, leaving Prince Andrei, fully understood everything that Natasha’s face told her. She no longer spoke to Natasha about the hope of saving his life. She alternated with her at his sofa and did not cry anymore, but prayed incessantly, turning her soul to that eternal, incomprehensible, whose presence was now so palpable over the dying man.

Prince Andrei not only knew that he would die, but he felt that he was dying, that he was already half dead. He experienced a consciousness of alienation from everything earthly and a joyful and strange lightness of being. He, without haste and without worry, awaited what lay ahead of him. That menacing, eternal, unknown and distant, the presence of which he never ceased to feel throughout his entire life, was now close to him and - due to the strange lightness of being that he experienced - almost understandable and felt.