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A function (or mapping) is called '''surjective''' if the image of its domain ([[range of values]]) coincides with its rang In other words, a function $ f : A \to B $ from a set $A$ to a set $B$ is

''of a differentiable function $f: \mathbb R^m \to \mathbb R^n$ with $n\leq m$''. A value $c\in \mathbb R^ differential of $f$ is surjective at every preimage of $c$. The concept can be readily extended to the case w

A function (or mapping) is called '''injective''' if distinct arguments have distinct In other words, a function $ f : A \to B $ from a set $A$ to a set $B$ is

A function (or mapping) is called '''bijective''' if it is both [[Injection|injective]] and [[Surjection|surjective]].

...ps any $y$ precisely into the element $x$ such that $f(x) =y$. The inverse function $g$ is then characterized by the following property: ...by $f^{-1}$ and hence it is commonly written $f^{-1} (y) = x$. The inverse function must not be confused with the ''preimage'' of $y$, which is usually denoted

...whose classes are the pre-images of the elements of $B$ (cf. [[Kernel of a function]]).

* The set $T(\Sigma)$ is [[Computable function|computational]]. ...ture and let $A$ be a $\Sigma$-algebra. As usual, the interpretation of a function symbol $f\in F$ in the algebra $A$ may be designated as $f^A$. Then the <i

...space $ X $, and if $ \chi $ is a continuous and uni-modular scalar-valued function on $ Y $, then ...uch that $ |f(x)| < \epsilon $ for $ x \in X \setminus K $). A uni-modular function $ \chi $ is one for which $ |\chi(y)| = 1 $ for all $ y \in Y $.

$#C+1 = 23 : ~/encyclopedia/old_files/data/S091/S.0901660 Symmetric function A function that does not change under any permutation of its independent variables. Th

$#C+1 = 39 : ~/encyclopedia/old_files/data/R080/R.0800260 Recursive function, ''partial recursive function''

A generalization of the notion of a [[Meromorphic function|meromorphic function]]. Let $ X $ is surjective and defines a bijective mapping of the set of irreducible components. If $

in each of which is defined a meromorphic function $ f _ \alpha $; compatibility condition). It is required to construct a function $ f $

is holomorphically convex if and only if it admits a proper surjective holomorphic mapping $ \phi $ ...x analytic spaces" , ''Contributions to function theory. Internat. Colloq. Function Theory, Bombay 1960'' , Tata Inst. (1960) pp. 1–15</TD></TR><TR><TD val

...$ as a $\Sigma$-number-algebra and $\alpha\colon C \longrightarrow A$ as a surjective $\Sigma$-algebra-morphism from $C$ onto $A$. Usually, $\alpha$ is chosen to Using a coordinatization $(C,\alpha)$, the notions of [[Computable function|computability]] (and other areas of theoretical computer science) can be tr

is surjective and has the property $ \chi ( 1) = 1 $. determines a function $ \widetilde{a} $

and is a far-reaching generalization of the ordinary exponential function regarded as a mapping of a straight line into itself. is a surjective mapping of $ M _ {p} $

be a surjective holomorphic mapping. Suppose this is an $ m $- is the different of the extension of algebraic function fields determined by $ f $.

...sion $v(x_1\leftarrow t_1,\ldots,x_n\leftarrow t_n)$ denotes an assignment function $v$ changed at variables $x_i$ to $t_i$. ...e|EM85}}. With help of these properties one can show for example that no (surjective) $\Sigma$-algebra-morphism exist between two given $\Sigma$-algebras {{Cite

An object $ (X,\pi,B) $, where $ \pi: X \to B $ is a continuous surjective mapping of a topological space $ X $ onto a topological space $ B $ (i.e., ...ld be called a morphism (respectively, an isomorphism) only if the induced function $ f: B \to B_{1} $ were continuous (respectively, a [[Homeomorphism|homeomo

($\langle \, .\, ,\, . \, \rangle$ being the scalar product) is the support function, maps this cone isomorphically into the space $C ( S ^ { n - 1 } )$ of cont ...ex bodies. Common examples are the mean width and the Steiner point $s$. A surjective mapping $\psi : \mathcal{K} ^ { n } \rightarrow \mathcal{K} ^ { n }$ with $

...by a tubular function (called "fonction tapis" in Thom's and "distance function" in Mather's terminology), and the fibrations and tubular functions associ ...arrow X$ is the fibre projection associated to $T _ { X }$ and the tubular function $\rho _ { X } : T _ { X } \rightarrow \mathbf{R}$ is a continuous mapping s

is either non-injective or non-surjective, then $ \lambda \in \sigma ( A) $. valued function $ R _ {A} ( \lambda ) = ( A- \lambda I ) ^ {-} 1 $,

...compact subsets of $\mathbf{C} ^ { n }$ (cf. also [[Entire function|Entire function]]; [[Uniform convergence|Uniform convergence]]). Let $\mathcal H ( \mathbf It is immediate to show that $\mathcal{F} \mu$ is an entire function. Moreover, since the exponentials are dense in $\mathcal{H} ( \mathbf{C} ^

...E = \{ z \in \mathbf C ^ { n } : \rho ( z ) < 0 \}$ is given by a defining function $\rho$ of class $C ^ { 2 }$, then one considers the quadratic form ...ly bounded domain $E$ is convex if and only if the Hessian of its defining function is positive semi-definite when restricted to the real tangent plane at any

...ace]] $\mathcal{H}$ may be viewed as a non-commutative generalization of a function algebra $C _ { 0 } ( \Omega )$ acting as multiplication operators on some $ ...niform convergence|uniform convergence]] on compact subsets of $\Omega$ in function algebras. Thus, it can be shown that $M ( A )$ is the strict completion of

A function $K ( x , y )$, $x , y \in E$, is called a reproducing kernel of such a Hilb i) for every fixed $y \in E$, the function $K ( x , y ) \in H$;

where the right hand-side is the usual partial derivative of the function $f\phi^{-1}:\mathbb{R}^n\rightarrow \mathbb{R}$ in the variables $x_1,\dots ...inates is given by $f(m)=\tilde{f}(x_1(m),\dots,x_n(m))$, then $Xf$ is the function given in local coordinates by the expression

is a finite surjective analytic mapping inducing an isomorphism of the open sets then any function that is holomorphic on $ U \setminus A $

...s that a function $f$ is given on $X$ (and also that the variable $y$ is a function of the variable $x$, or that $y$ depends on $x$) and one writes $f:X\to Y$. ...r finally a dependence can exist and remain unknown" . The definition of a function as a correspondence between two arbitrary sets (not necessarily consisting

and surjective if $ \mathop{\rm Ran} R = B $. is functional, that is, it is the graph of a function from $ A $

is a class function on $ G: \chi _ {V} \in { \mathop{\rm CF} } ( G,K ) $. 2) A class function $ \varphi \in { \mathop{\rm CF} } ( G, \mathbf C ) $

the Poincaré problem, i.e. can any meromorphic function be represented in the form $ f/g $, pseudoconvex function exhausting it.

...ecause after all a (partial) differential equation is a relation between a function, its dependent variables and its derivatives up to a certain order. In the is surjective for all $ m \in M $.

...] in $s$. Langlands' general theory of Eisenstein series implies that this function has a meromorphic continuation and hence one can "evaluate" at $s = 0$. H ...es certain $L$-functions $L ( \omega , r , s )$ (cf. also [[L-function|$L$-function]]) and the answer to the above questions depends on the behaviour of certai

...rue for the columns of $A$.) The determinant of $A$ can be considered as a function of its rows: 1) $d$ is a linear function of any row of $A$:

...by H.R. Fischer [[#References|[a8]]] (1959). With respect to the study of function spaces, limit spaces are more convenient than topological spaces. A more re ...|epimorphism]]; [[Bimorphism|bimorphism]]) if and only if it is injective (surjective, bijective).

...normal scheme and let $f$ be a rational (meromorphic in the analytic case) function on $X$. A principal Weil divisor is defined canonically: ...he zeros, while $\sum n_W^- W$ is known as the divisor of the poles of the function $f$. The set of principal Weil divisors is a subgroup $Z_p^1(X)$ of the gro

is the required function and $ f ( t) $ the given function with values in a complex Banach space $ E $;

...of Abelian varieties, some results involving Drinfel'd modules over global function fields $L$ can be proved, whose analogues over number fields $L$ are far fr [[L-function|$L$-function]]). The invention and basic theory as well as large parts of the deeper res

...gic) and the property that every epimorphism (in the categorical sense) is surjective; ...mathbf{B}$ be algebras and $h : \mathbf{A} \twoheadrightarrow \mathbf B$ a surjective homomorphism. Finally, let $F _ { 0 }$ be the smallest $\mathcal{D}$-filter

...njunction, some people now (1998) prefer to study the lambda-calculus with surjective pairing, which turns out to be equivalent to a Cartesian closed category, s ...ive functions (cf. also [[Primitive recursive function|Primitive recursive function]]).

...he theory of generalized functions (cf. [[Generalized function|Generalized function]]). It is interesting that there is no norm on $ S $ is continuous (respectively, if every surjective continuous linear mapping from $ E _ {2} $

...\rightarrow \operatorname{Sets}$ is the semantical meaning (or denotation) function of $\mathcal{L}$, and $\vdash$ is the syntactical provability relation of $ ...le, fails to have almost any form of the amalgamation property and has non-surjective epimorphisms. Almost all of these theorems remain true if one throws away t

...the same regularity. This follows from the [[Implicit function|Implicit function theorem]]. ...cases: where $m\le n$ and $M$ is injective, and where $m\ge n$ and $M$ is surjective. The conclusion reduces the classification of nonlinear germs to that of l

is a function assigning to each object $ A $ sends each topological space to its underlying set, and each continuous function to itself ( "forgetting" the continuity). Similarly, one has forgetful func