Continuation (math)

The continuation of a figure is a term from mathematics that is used in particular in analysis and topology . A continuation of a mapping is understood to mean a further mapping that corresponds to the given mapping on a subset of its domain . It is of particular interest whether there are continuations to continuous or analytical functions that are also continuous or analytical.

definition

Be and set . A mapping is called continuation of the mapping if and only if is a subset of and holds for all . ${\displaystyle X,\,Y}$ ${\displaystyle A}$ ${\displaystyle f\colon X\to Y}$ ${\displaystyle g\colon A\to Y}$ ${\displaystyle A}$ ${\displaystyle X}$ ${\displaystyle g(x)=f(x)}$ ${\displaystyle x\in A}$

Continuous continuation

definition

Let and be topological spaces , a subspace of and a continuous mapping . Analogous to the definition above, a mapping is called continuous continuation of , if is continuous and holds for all . ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle A\subset X}$ ${\displaystyle X}$ ${\displaystyle g\colon A\to Y}$ ${\displaystyle f\colon X\to Y}$ ${\displaystyle g}$ ${\displaystyle f}$ ${\displaystyle g(x)=f(x)}$ ${\displaystyle x\in A}$

Examples

• The function , defined by , is continuous on its domain and has a continuous continuation on whole , which is ${\displaystyle g\colon \mathbb {R} \setminus \{0\}\to \mathbb {R} }$ ${\displaystyle x\mapsto {\tfrac {x}{x}}+5x}$ ${\displaystyle \mathbb {R} \setminus \{0\}}$ ${\displaystyle \mathbb {R} }$

${\displaystyle f(x)={\begin{cases}{\frac {x}{x}}+5x&\mathrm {f{\ddot {u}}r} \ x\in \mathbb {R} \setminus \{0\},\\1&\mathrm {f{\ddot {u}}r} \ x=0\,.\end{cases}}}$

Here the function is continued on a further point and in this special case one speaks of a definition gap that can be continuously remedied .

• The function , defined by , is continuously on its domain and has a continuous continuation on the whole . Because according to the rule of de l'Hospital, it is , and thus is ${\displaystyle g\colon \mathbb {R} \setminus \{0\}\to \mathbb {R} }$ ${\displaystyle x\mapsto {\tfrac {1}{x}}\sin(x)}$ ${\displaystyle \mathbb {R} \setminus \{0\}}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \textstyle \lim _{x\to 0}{\tfrac {1}{x}}\sin(x)=1}$

${\displaystyle f(x)={\begin{cases}{\frac {1}{x}}\sin(x)&\mathrm {f{\ddot {u}}r} \ x\in \mathbb {R} \setminus \{0\},\\1&\mathrm {f{\ddot {u}}r} \ x=0\,\end{cases}}}$

a steady continuation of . ${\displaystyle g}$

• The function , defined by , is continuous in its domain , but unlike the functions mentioned above, it does not have a continuous continuation over the entire number range , since the limit value does not exist. ${\displaystyle g\colon \mathbb {R} \setminus \{0\}\to \mathbb {R} }$ ${\displaystyle x\mapsto \sin({\tfrac {1}{x}})}$ ${\displaystyle \mathbb {R} \setminus \{0\}}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \textstyle \lim _{x\to 0}\sin({\tfrac {1}{x}})}$
• In the mathematical area of functional analysis , the Fourier transformation is considered. This is a picture on the Schwartz room . Since the Schwartz space lies close to the space of the square integrable functions , the Fourier transformation can be continued continuously on . However, in this space it no longer has the usual integral representation that it has in the Schwartz space. ${\displaystyle {\mathcal {F}}\colon {\mathcal {S}}\to {\mathcal {S}}}$ ${\displaystyle L^{2}}$ ${\displaystyle L^{2}}$

Continuation of Tietze's sentence

→ Main article : Tietze's continuation

Tietze's continuation theorem characterizes topological spaces in which continuous functions can always be continued continuously on closed subsets . It is exactly the normal topological spaces in which this is always possible. The theorem can be understood as a generalization of Urysohn's lemma . A consequence of Tietze's continuation theorem is the continuation lemma .

Lipschitz continuous functions

Continuous mappings , whereby , can have the stronger property of Lipschitz continuity . Therefore the question arises whether one can choose the continuous continuations in such a way that the Lipschitz continuity is preserved. The set of Kirszbraun says that this even with preserving the Lipschitz constant is possible. The lemma McShane expands this statement made to more general classes of area. ${\displaystyle U\rightarrow \mathbb {R} ^{m}}$ ${\displaystyle U\subset \mathbb {R} ^{n}}$

Periodic continuation

→ Main article : Periodic continuation

Another possibility to continue a function systematically is the periodic continuation. A function defined on a limited interval is continued in such a way that its function values ​​are repeated cyclically outside the starting interval with a fixed interval. Such a function is called periodic .

Restriction

→ Main article : Limitation

The opposite concept to the continuation of functions is the restriction of the domain of an image.

See also

• Analytical continuation
• Continuation of Dugundji

Individual evidence

1. ↑ Continuation of an illustration. In: Guido Walz (Ed.): Lexicon of Mathematics. Volume 1: A to Eif. Spectrum - Akademischer Verlag, Heidelberg et al. 2000, ISBN 3-8274-0303-0 .
2. ↑ Dušan Repovš, Pavel Semenov Vladimirovič: Continuous selections of multivalued mappings . In: Mathematics and its Applications . tape 455 . Kluwer Academic, Dordrecht et al. 1998, ISBN 0-7923-5277-7 , p. 23-24 .