Madhuri Dixit

In Mathematical, the together of definition D _F of a function F whose starting together is noted E and the Together of arrival F , is the whole of the previous of F , i.e. the whole of the elements of E that F puts in relation to elements of F ; it is thus the whole of the elements X of E for which F ( X ) exists:

D_f = \ {X \ in E \ |\, \ exists \ there \ in F \,/\, there = F (X) \} \,

D _F is still called field of definition of F or field of F .

One should not confuse the field of definition of a function F (for memory: D _F ) with its starting whole (for memory: E ). It happens however that both are equal: the function is then a application . It is known as in this case well defined or everywhere definite in E .

As counterexample, let us consider the function $F: \ begin {pmatrix} \ mathbb {R} \ longrightarrow \ mathbb {R} \ \ X \ longmapsto \ frac {1} {X} \ end {pmatrix} \,$ .

This function is not defined into 0: “ F (0)” does not exist.

The whole of definition of this function is thus $\ mathbb {R} ^* \,$ (recall: $\ mathbb {R} ^* = \ mathbb {R} - \ {0 \} \,$ ). It differs from starting sound together, $\ mathbb {R} \,$ ; this function is thus not an application.

However, it is always possible to transform a function into application, for example in the restricting with its field of definition. This restriction is usually noted “ $F |_ {D_f} \,$ ”. It is an application by construction.

Thus, in our example, the function $F |_ {\ mathbb {R} ^*}: \ begin {pmatrix} \ mathbb {R} ^* \ longrightarrow \ mathbb {R} ^* \ \ X \ longmapsto \ frac {1} {X} \ end {pmatrix} \,$ is well an application.

Another solution to transform a function into application consists with the to prolong , i.e. to choose an image in the whole of arrival for each element without image of the starting whole. In particular, if a numerical function F is not defined in a point X ₀, it is possible to prolong it in this point by replacing it by another function, called prolongation of F in X ₀ and usually noted “ $\ bar F \,$ ”, and such as:

the prolongation of F is equal to F on D _F :

\ forall \ X \ in D_f \, \ bar F (X) = F (X) \,

the prolongation of F has a defined value, has , at the point X ₀:

\ exists \ has \ in F/\, \ bar F (x_0) = has \,

Thus, in our example, one can transform the function $F \,$ in application by prolonging it at the origin by: $F (0) = 0 \,$

Note: enough often, to reduce the notations, the prolongation is noted same manner as the initial function. This ambiguity is without consequence if the prolongation is clarified and replaces at once and definitively the initial function.

See too

Internal bonds

External bonds

Theory and exercises on the fields of definitions

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