Definition gap

In the mathematical subfield of analysis , a function has definition gaps if individual points are excluded from its domain . Usually it is about real , continuous or differentiable functions. The definition gaps are the places where one would have to divide by zero or the like, for example in the case of fractional rational functions . The definition gaps of a function can be classified and, if necessary, "repaired" so that the function can be continued there with the desired properties. In this case, the function can be continued continuously and has continuously resolvable definition gaps .

In particular, if a definition gap cannot be continuously removed, for example because the function there tends towards infinity or oscillates very quickly, the gap is also referred to as a singularity , although the linguistic usage in these cases is not always uniform. Often definition gap and singularity are used as synonyms.

In the case of complex-valued functions that are holomorphic in the vicinity of a definition gap , one speaks of isolated singularities . There the classification is simpler and there are far-reaching statements for which there are no equivalents in real functions.

definition

Function with definition gap ${\displaystyle x_{0}}$

Let be an interval , a point from the inside of the interval and a superset of . A continuous function that is defined everywhere on the superset except at the point has a definition gap. ${\displaystyle I=[a,b]\subset \mathbb {R} }$ ${\displaystyle x_{0}\in \left]a,b\right[}$ ${\displaystyle O}$ ${\displaystyle I}$ ${\displaystyle f\colon O\setminus \{x_{0}\}\to \mathbb {R} }$ ${\displaystyle O}$ ${\displaystyle x_{0}}$ ${\displaystyle x_{0}}$

Gaps in definition that can be continuously removed

Let be a definition gap of the continuous function . If there is a continuous function with for all , then is a continuous continuation of . The definition gap is then continuously liftable or continuously remedied and the function is called continuously supplementable or continuously continued . ${\displaystyle x_{0}}$ ${\displaystyle f\colon I\setminus \{x_{0}\}\to \mathbb {R} }$ ${\displaystyle {\tilde {f}}\colon I\to \mathbb {R} }$ ${\displaystyle {\tilde {f}}(x)=f(x)}$ ${\displaystyle x\in I\setminus \{x_{0}\}}$ ${\displaystyle {\tilde {f}}}$ ${\displaystyle f}$ ${\displaystyle f}$

Does the limit exist

${\displaystyle \lim _{x\to x_{0}}f(x)=:r\,,}$

then there is a gap in the definition of . In this case, through ${\displaystyle x_{0}}$ ${\displaystyle f}$

${\displaystyle {\tilde {f}}(x):={\begin{cases}f(x),&x\in I\setminus \{x_{0}\}\\r,&x=x_{0}\end{cases}}}$

a continuous continuation of defined without a definition gap. ${\displaystyle {\tilde {f}}}$ ${\displaystyle f}$

Properties of continuous continuations

• If there is a continuous continuation, then it is unique because the limit

${\displaystyle {\tilde {f}}(x_{0})=\lim _{x\to x_{0} \atop x\in I}f(x)}$

is unique.

• From this follows the criterion: it can be continued continuously if and only if the limit value exists. ${\displaystyle f}$ ${\displaystyle x_{0}}$ ${\displaystyle \textstyle \lim _{x\to x_{0}}f(x)}$
• When a feature is represented as a fraction whose numerator and denominator function on a common root differentiable are, the true rule of de l'Hospital : ${\displaystyle x_{0}}$

${\displaystyle \lim _{x\to x_{0}}{\frac {u(x)}{v(x)}}=\lim _{x\to x_{0}}{\frac {u'(x)}{v'(x)}}.}$

• The constriction theorem offers a more general way of finding a steady continuation . It also applies to non-continuous functions.
• A continuation is always continuous, but may not be differentiable. The amount function is differentiable but cannot be continued to zero. Even if a continuation is smooth , it doesn't have to be analytical . ${\displaystyle \mathbb {R} \setminus \{0\}}$
• In the complex, further statements apply due to the properties of holomorphic functions : A continuous continuation is already an analytical continuation . The Riemann Hebbarkeitssatz states that the definition gap of a holomorphic function is already liftable when the function in a suitable environment as defined gap limits is. In the real world, no comparable statement applies; there could also be a non liftable discontinuity are present.

Other types of definition gaps

In addition to the definition gaps that can be continuously removed, there are also various types of jump points as well as poles and essential singularities . Functions with such definition gaps cannot be continued continuously.

Examples

• The function is continuous in its entire domain , but has a definition gap at position 0. This is a pole. ${\displaystyle f(x)={\tfrac {1}{x}}}$ ${\displaystyle D=\mathbb {R} \setminus \{0\}}$
• Be given

${\displaystyle f\colon \mathbb {R} _{\geq 0}\setminus \{1\}\to \mathbb {R} ,\,x\mapsto {\frac {{\sqrt {x}}-1}{x-1}}\,.}$

The function can be continued continuously because the following applies to the limit value ${\displaystyle f}$ ${\displaystyle x_{0}=1}$

${\displaystyle \lim _{x\to 1}{\frac {{\sqrt {x}}-1}{x-1}}=\lim _{x\to 1}{\frac {{\sqrt {x}}-1}{({\sqrt {x}}+1)({\sqrt {x}}-1)}}=\lim _{x\to 1}{\frac {1}{{\sqrt {x}}+1}}={\frac {1}{2}}}$

and thus the continuation is

${\displaystyle {\tilde {f}}\colon \mathbb {R} _{\geq 0}\to \mathbb {R} ,\,x\mapsto {\begin{cases}{\frac {{\sqrt {x}}-1}{x-1}}&,x\neq 1\\{\frac {1}{2}}&,x=1\,.\end{cases}}}$.

In this example one can also notice that it is also possible to write without differentiating between cases, because it applies to everyone . ${\displaystyle {\tilde {f}}}$ ${\displaystyle {\tilde {f}}(x)={\tfrac {1}{{\sqrt {x}}+1}}}$ ${\displaystyle x\geq 0}$

• In other cases it may be that the case distinction is inevitable. So has about

${\displaystyle g\colon \mathbb {R} \setminus \{0\}\to \mathbb {R} ,\,x\mapsto x\cdot \sin({\tfrac {1}{x}})}$

the steady continuation

${\displaystyle {\tilde {g}}\colon \mathbb {R} \to \mathbb {R} ,\,x\mapsto {\begin{cases}x\cdot \sin({\tfrac {1}{x}})&,x\neq 0\\0&,x=0\end{cases}}}$.

Fractional rational functions

The quotient is a fractional rational function

${\displaystyle f(x)={\frac {u(x)}{v(x)}}}$

from two completely rational functions and . ${\displaystyle u}$ ${\displaystyle v}$

A fractional rational function has a definition gap if and only if the rational function has a zero in the denominator . Functions of this special class can only have poles or continuously resolvable definition gaps as definition gaps.

The definition gap can only be continuously removed if the completely rational functions in the denominator and numerator have a zero in the same place. For the completely rational functions and the behavior at the zeros is known: ${\displaystyle u}$ ${\displaystyle v}$

The zeros of the numerator and denominator functions can be factored out. So if and have a zero at that point , it is always ${\displaystyle u(x)}$ ${\displaystyle v(x)}$ ${\displaystyle x_{0}}$

${\displaystyle u(x)=(x-x_{0})^{N_{u}}\;s(x)}$

and

${\displaystyle v(x)=(x-x_{0})^{N_{v}}\;t(x)}$

in which

${\displaystyle s(x_{0})\neq 0\land t(x_{0})\neq 0}$.

The natural numbers and are also referred to as the order (or multiplicity) of the respective zero. ${\displaystyle N_{u}}$ ${\displaystyle N_{v}}$

Obviously one can reduce the common factors of the zeros (at least for ). The result of the cut is the only candidate for steady continuation after . ${\displaystyle x\neq x_{0}}$ ${\displaystyle x_{0}}$

• If so , then there is a definition gap that can be continuously corrected, whereby the limit value is given by 0. ${\displaystyle N_{u}>N_{v}>0}$
• If so , then there is a definition gap that can be continuously corrected, whereby the limit value is given by . ${\displaystyle N_{u}=N_{v}>0}$ ${\displaystyle s(x_{0})/t(x_{0})}$
• If so , then there is a pole. ${\displaystyle N_{u}<N_{v}}$

example

The function

${\displaystyle f(x)={\frac {x^{3}+4x^{2}+5x+2}{x^{3}+x^{2}-x-1}}={\frac {(x+1)^{2}(x+2)}{(x+1)^{2}(x-1)}}}$

has for a gap that can be eliminated by truncating with the value , which makes the function ${\displaystyle x=-1}$ ${\displaystyle (x+1)^{2}}$

${\displaystyle {\tilde {f}}(x)={\frac {x+2}{x-1}}}$

as well as with continuous continuation. It's mind as well as for undefined, there is a pole in front. ${\displaystyle x=-1}$ ${\displaystyle {\tilde {f}}}$ ${\displaystyle f}$ ${\displaystyle x=+1}$

An example to illustrate the distinction between a pole and a definable gap in the definition. The function

${\displaystyle f(x)={\frac {x-1}{x^{2}-2x+1}}={\frac {(x-1)}{(x-1)^{2}}}}$

has for a definition gap that can be reduced with the value on the function ${\displaystyle x=+1}$ ${\displaystyle (x-1)}$

${\displaystyle {\tilde {f}}(x)={\frac {1}{x-1}}}$ leads.

Since is as well as for undefined, the shortening did not resolve the gap. Therefore, there is a pole position and not a definable gap in the definition. ${\displaystyle {\tilde {f}}}$ ${\displaystyle f}$ ${\displaystyle x=+1}$ ${\displaystyle x=+1}$

See also

• Point of discontinuity

Individual evidence

1. ↑ cf. Harald Scheid / Wolfgang Schwarz: Elements of linear algebra and analysis . Spektrum, Akad. Verl., Heidelberg 2009, ISBN 978-3-8274-1971-2 , pp. 237 .