What Makes A Function Undefined

Expression which is not assigned an estimation

In mathematics, the term undefined is often used to refer to an expression which is not assigned an interpretation or a value (such as an indeterminate form, which has the propensity of assuming different values).^[ane] The term can take on several different meanings depending on the context. For instance:

Undefined terms [edit]

In ancient times, geometers attempted to ascertain every term. For example, Euclid defined a betoken equally "that which has no part". In modern times, mathematicians recognize that attempting to ascertain every give-and-take inevitably leads to circular definitions, and therefore leave some terms (such as "point") undefined (meet archaic notion for more than).

This more abstract approach allows for fruitful generalizations. In topology, a topological space may exist defined as a set of points endowed with certain backdrop, but in the full general setting, the nature of these "points" is left entirely undefined. Likewise, in category theory, a category consists of "objects" and "arrows", which are again primitive, undefined terms. This allows such abstruse mathematical theories to be practical to very diverse physical situations.

In arithmetic [edit]

The expression 0 / 0 is undefined in arithmetic, as explained in segmentation by zero (the aforementioned expression is used in calculus to stand for an indeterminate grade).

Mathematicians have dissimilar opinions as to whether 0⁰ should be divers to equal i, or be left undefined.

Values for which functions are undefined [edit]

The set of numbers for which a function is defined is chosen the domain of the part. If a number is non in the domain of a function, the function is said to exist "undefined" for that number. Two common examples are ${\textstyle f(x)={\frac {one}{x}}}$ , which is undefined for ${\displaystyle x=0}$ , and ${\displaystyle f(x)={\sqrt {x}}}$ , which is undefined (in the real number system) for negative ${\displaystyle x}$ .

In trigonometry [edit]

In trigonometry, the functions ${\displaystyle \tan \theta }$ and ${\displaystyle \sec \theta }$ are undefined for all ${\textstyle \theta =180^{\circ }\left(n-{\frac {ane}{2}}\correct)}$ , while the functions ${\displaystyle \cot \theta }$ and ${\displaystyle \csc \theta }$ are undefined for all ${\displaystyle \theta =180^{\circ }(n)}$ .

In computer science [edit]

Notation using ↓ and ↑ [edit]

In computability theory, if ${\displaystyle f}$ is a fractional part on ${\displaystyle S}$ and ${\displaystyle a}$ is an element of ${\displaystyle S}$ , then this is written as ${\displaystyle f(a)\downarrow }$ , and is read as "f(a) is defined."^[3]

If ${\displaystyle a}$ is not in the domain of ${\displaystyle f}$ , then this is written as ${\displaystyle f(a)\uparrow }$ , and is read as " ${\displaystyle f(a)}$ is undefined".

The symbols of infinity [edit]

In analysis, measure theory and other mathematical disciplines, the symbol ${\displaystyle \infty }$ is frequently used to announce an infinite pseudo-number, along with its negative, ${\displaystyle -\infty }$ . The symbol has no well-defined meaning by itself, just an expression like ${\displaystyle \left\{a_{north}\right\}\rightarrow \infty }$ is autograph for a divergent sequence, which at some indicate is eventually larger than any given real number.

Performing standard arithmetic operations with the symbols ${\displaystyle \pm \infty }$ is undefined. Some extensions, though, define the following conventions of addition and multiplication:

No sensible extension of addition and multiplication with ${\displaystyle \infty }$ exists in the following cases:

For more detail, see extended real number line.

Singularities in complex analysis [edit]

In circuitous assay, a signal ${\displaystyle z\in \mathbb {C} }$ where a holomorphic function is undefined is called a singularity. One distinguishes between removable singularities (i.east., the office can be extended holomorphically to ${\displaystyle z}$ ), poles (i.e., the part can be extended meromorphically to ${\displaystyle z}$ ), and essential singularities (i.due east., no meromorphic extension to ${\displaystyle z}$ can exist).

References [edit]

1. ^ Weisstein, Eric West. "Undefined". mathworld.wolfram.com . Retrieved 2019-12-15 .
2. ^ "Undefined vs Indeterminate in Mathematics". world wide web.cut-the-knot.org . Retrieved 2019-12-15 .
3. ^ Enderton, Herbert B. (2011). Computability: An Introduction to Recursion Theory. Elseveier. pp. 3–vi. ISBN978-0-12-384958-8.

Farther reading [edit]

• Smart, James R. (1988). Modern Geometries (Tertiary ed.). Brooks/Cole. ISBN0-534-08310-two.

What Makes A Function Undefined,

Source: https://en.wikipedia.org/wiki/Undefined_%28mathematics%29

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