Smooth function

Smoothness is a redirect to this article. For the concept of number theory see smooth number .

A smooth function is a mathematical function that is infinitely differentiable (especially continuous ). The designation “smooth” is motivated by the observation: The graph of a smooth function has no “corners”, ie places where it cannot be differentiated. This makes the graph look “particularly smooth” everywhere. For example, every holomorphic function is also a smooth function. In addition, smooth functions are used as truncation functions or as test functions for distributions .

definition

Conventions

For a non-empty open subset of the set of one calls real-valued and completely continuous functions with , or . Correspondingly, the set of functions that are once continuously differentiable is denoted by and for each natural number the set of functions that are continuously differentiable is denoted by. ${\displaystyle D\subset \mathbb {R} }$ ${\displaystyle D}$ ${\displaystyle C(D)}$ ${\displaystyle C^{0}(D)}$ ${\displaystyle C^{0}(D,\mathbb {R} )}$ ${\displaystyle C^{1}(D)}$ ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle C^{n}(D)}$

The set of the -time continuously differentiable function is recursive by ${\displaystyle n}$

${\displaystyle f\in C^{n}(D)\Leftrightarrow f\in C^{1}(D){\text{ und }}f'\in C^{n-1}(D)}$

Are defined. It always applies

${\displaystyle C^{n}(D)\subset C^{n-1}(D)\subset \dotsb \subset C^{1}(D)\subset C^{0}(D)}$.

Smooth functions

A function is called infinitely often (continuously) differentiable or smooth if applies to all . The set of all smooth functions is noted with and it applies ${\displaystyle f\colon D\to \mathbb {R} }$ ${\displaystyle f\in C^{n}(D)}$ ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle D}$ ${\displaystyle C^{\infty }(D)}$

${\displaystyle C^{\infty }(D):=\bigcap _{n\in \mathbb {N} }C^{n}(D).}$

This description is particularly useful for topological considerations.

Generalizations

The concept of smooth function can be generalized to more general cases without difficulty . It is said that a function is infinitely differentiable or smooth if all partial derivatives are infinitely differentiable. Smooth functions between smooth manifolds are also defined and examined. ${\displaystyle f\colon \mathbb {R} ^{m}\supset D\to \mathbb {R} ^{n}}$

properties

• All differentiable derivatives are necessarily continuous, since differentiability implies continuity.
• The term “sufficiently smooth” is often found in mathematical considerations. By this is meant that the function for a sufficiently large in lies, so just is differentiable, to carry out the current line of thought. This is formulated in such a way as to avoid too strong (and not sensible) restriction by “infinitely often differentiable” and, on the other hand, not to have to go through all the prerequisites that are already met in the cases usually considered, or if the exact ones Restriction for other reasons does not play a role: As a theoretical argument it can be stated that for all the -fold differentiable and also the infinitely often differentiable functions and the analytical functions with regard to many common metrics lie close to the continuous. If, for example, there is a physical problem in which small changes are not important, there are functions that are arbitrarily “close” to a continuous function under consideration and that fulfill the mathematical conditions set; it may even be possible to show that the property that has been proven for certain functions is transferred to a larger area in which they are located close together. If it can be seen from the context that only sufficiently smooth functions are considered (e.g. by specifying the degree of differentiability), the addition “sufficient” is sometimes omitted. ${\displaystyle n}$ ${\displaystyle C^{n}(D)}$ ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle n}$
• In addition, one denotes the set of all analytical functions , these are the infinitely often differentiable functions, the Taylor expansion of which converges towards the given function around any point in a neighborhood. It is then noteworthy that each of the following inclusions is real in the real-valued case. In the case of complex-valued and complex differentiable, or rather holomorphic, functions , every function that is complexly differentiable on an open set is infinitely differentiable and even analytical. That is why the differentiability mostly relates to functions whose definition and target set are the real numbers, vector spaces or manifolds over the real numbers or the like. ${\displaystyle C^{\omega }(D)}$ ${\displaystyle C^{0}(D)\supset \dotsb \supset C^{n}(D)\supset C^{n+1}(D)\supset \dotsb \supset C^{\infty }(D)\supset C^{\omega }(D)}$ ${\displaystyle C^{n}(D)}$
• Each and also (as well as ) is an (infinitely dimensional) vector space . ${\displaystyle C^{n}(D)}$ ${\displaystyle C^{\infty }(D)}$ ${\displaystyle C^{\omega }(D)}$

Examples

• All polynomial functions are infinitely differentiable and even analytical.
• By

${\displaystyle f\colon \mathbb {R} \to \mathbb {R} ,\;\;\;x\mapsto {\begin{cases}x^{n}&\mathrm {f{\ddot {u}}r} \quad x\geq 0\\-x^{n}&\mathrm {f{\ddot {u}}r} \quad x<0\end{cases}}}$

defined function is times continuously differentiable ( ), the th derivation is, however, at the place not continuously differentiable, ie . ${\displaystyle (n-1)}$ ${\displaystyle f\in C^{n-1}(\mathbb {R} )}$ ${\displaystyle (n-1)}$ ${\displaystyle f^{(n-1)}(x)=n!\,\left|x\right|}$ ${\displaystyle x=0}$ ${\displaystyle f\notin C^{n}(\mathbb {R} )}$

• The function

${\displaystyle g\colon \mathbb {R} \to \mathbb {R} ,\;\;\;x\mapsto {\begin{cases}\mathrm {e} ^{-{\frac {1}{x^{2}}}}&\mathrm {f{\ddot {u}}r} \quad x\neq 0\\0&\mathrm {f{\ddot {u}}r} \quad x=0\end{cases}}}$

is an infinitely often differentiable function, but not an analytical function, because the Taylor series around the zero point does not coincide with the function in any neighborhood around 0, since all derivatives at 0 assume the value 0.

• But it is also the same

${\displaystyle h\colon \mathbb {R} \to \mathbb {R} ,\;\;\;x\mapsto {\begin{cases}\mathrm {e} ^{-{\frac {1}{x^{2}}}}&\mathrm {f{\ddot {u}}r} \quad x>0\\0&\mathrm {f{\ddot {u}}r} \quad x\leq 0\end{cases}}}$

infinitely differentiable. Obviously no global statements can be derived from local knowledge of an infinitely differentiable function (here, for example, applies to all positive , but nevertheless ). ${\displaystyle g(x)=h(x)}$ ${\displaystyle x}$ ${\displaystyle g\neq h}$

• The Schwartz space contains only smooth functions and is a real subset of the infinitely often differentiable functions.

application

These last two examples are important tools for constructing examples of smooth functions with special properties. In the following way one can construct a smooth decomposition of the one (here: of ): ${\displaystyle \mathbb {R} }$

• The function is infinitely differentiable with a compact carrier . ${\displaystyle j\colon \mathbb {R} \to \mathbb {R} ,\;x\mapsto h(1+x)\cdot h(1-x)}$ ${\displaystyle [-1,1]}$

• The function

${\displaystyle k\colon \mathbb {R} \to \mathbb {R} ,\;x\mapsto {\frac {h(1+x)}{h(1+x)+h(1-x)}}}$

is infinitely differentiable and the following applies:

${\displaystyle {\begin{array}{ccc}k(x)=0&\mathrm {f{\ddot {u}}r} &x\leq -1\\0<k(x)<1&\mathrm {f{\ddot {u}}r} &-1<x<1\\k(x)=1&\mathrm {f{\ddot {u}}r} &x\geq 1\end{array}}}$

Topologization

Be an open subset . In the space of smooth functions , a topology is explained in particular in distribution theory. The family of semi-norms ${\displaystyle D\subset \mathbb {R} ^{n}}$ ${\displaystyle f\colon D\to \mathbb {R} }$

${\displaystyle f\in C^{\infty }(D)\mapsto \sum _{|\alpha |=m}\sup _{x\in K}\left|{\frac {\partial ^{\alpha }}{\partial x^{\alpha }}}f(x)\right|}$

with and goes through all compacts, turning the space of smooth functions into a locally convex space . This is complete and therefore a Fréchet room . In addition, since each closed and bounded set is compact , this is even a Montel space . The space of the smooth functions together with this locally convex topology is usually referred to as. ${\displaystyle m\in \mathbb {N} }$ ${\displaystyle K\subset D}$ ${\displaystyle C^{\infty }(D)}$ ${\displaystyle {\mathcal {E}}(D)}$

See also

• Differentiation class

literature

• Otto Forster: Analysis 2. Differential calculus in R ⁿ . Ordinary differential equations. Vieweg-Verlag, 7th edition 2006, ISBN 3-528-47231-6 .