hcju/proofwikips huggingface dataset:
Definition:Half-Range Fourier Sine Series/Range Pi
Let $\map f x$ be a real function defined on the interval $\openint 0 \pi$. Then the ''' half-range Fourier sine series''' of $\map f x$ over $\openint 0 \pi$ is the series: :$\map f x \sim \displaystyle \sum_{n \mathop = 1}^\infty b_n \sin n x$ where for all $n \in \Z_{> 0}$: :$b_n = \displaystyle \frac 2 \pi \int_0^\pi \map f x \sin n x \rd x$
Definition:Everywhere Dense/Normed Vector Space
Let $M = \struct {X, \norm {\, \cdot \,}}$ be a normed vector space. Let $Y \subseteq X$ be a subset of $X$. Suppose: :$\forall x \in X : \forall \epsilon \in \R : \epsilon > 0 : \exists y \in Y : \norm {x - y} < \epsilon$ Then $Y$ is '''(everywhere) dense''' in $X$.
Definition:Separable Space/Normed Vector Space
Let $M = \struct {X, \norm {\, \cdot \,}}$ be a normed vector space. Let $Y \subseteq X$ be a subset of $X$. Let $Y$ be dense in $X$. Let $Y$ be a countable set: :$Y = \set {y_1, y_2, \ldots}$ Suppose: :$\forall x \in X : \forall \epsilon \in \R : \epsilon > 0 : \exists y_{n \mathop \in \N} \in Y : \norm {y_n - x} < \epsilon$ Then $X$ is '''separable'''.
Definition:Conjugate Point/Definition 3
Let $y = \map y x$ and $y = \map {\tilde y} x$ be extremal functions. Let: :$M = \paren {a, \map y a}$ :$\tilde M = \paren {\tilde a, \map y {\tilde a} }$ Let both $y = \map y x$ and $y = \map {\tilde y} x$ pass through the point $M$. Let  :$\displaystyle \lim_{\norm {\map y x - \map {\tilde y} x}_{1, \infty} \to 0} \sqbrk {\paren {x, \map y x}: \map y x - \map {\tilde y} x = 0} = \tilde M$ In other words, let $\tilde M$ be the limit points of intersection of $y = \map y x$ and $y = \map {\tilde y} x$ as $\norm {\map y x - \map {\tilde y} x}_{1, \infty} \to 0$. {{explain|the notation $\norm {\map y x - \map {\tilde y} x}_{1, \infty}$ with particular regard to the subscript This is most likely the standard norm of Sobolev space, also know as np-norm. Original source never calls it by word; only by special symbols}} Then $\tilde M$ is '''conjugate''' to $M$.
Definition:Separated by Neighborhoods/Sets/Neighborhoods
Let $A, B \subseteq S$ such that: :$\exists N_A, N_B \subseteq S: \exists U, V \in \tau: A \subseteq U \subseteq N_A, B \subseteq V \subseteq N_B: N_A \cap N_B = \O$ That is, that $A$ and $B$ both have neighborhoods in $T$ which are disjoint. Then $A$ and $B$ are described as '''separated by neighborhoods'''.
Definition:Separated by Neighborhoods/Points/Neighborhoods
Let $x, y \in S$ such that: :$\exists N_x, N_y \subseteq S: \exists U, V \in \tau: x \in U \subseteq N_x, y \in V \subseteq N_y: N_x \cap N_y = \O$ That is, that $x$ and $y$ both have neighborhoods in $T$ which are disjoint. Then $x$ and $y$ are described as '''separated by neighborhoods'''.
Definition:Separated by Neighborhoods/Points/Open Sets
Let $x, y \in S$ such that: :$\exists U, V \in \tau: x \in U, y \in V: U \cap V = \O$ That is, that $x$ and $y$ both have open neighborhoods in $T$ which are disjoint. Then $x$ and $y$ are described as '''separated by (open) neighborhoods'''.
Definition:Separated by Neighborhoods/Sets/Open Sets
Let $A, B \subseteq S$ such that: :$\exists U, V \in \tau: A \subseteq U, B \subseteq V: U \cap V = \O$ That is, that $A$ and $B$ both have open neighborhoods in $T$ which are disjoint. Then $A$ and $B$ are described as '''separated by (open) neighborhoods'''.
Definition:Positive Definite Functional
Let $f : S \to T$ be a mapping. Let $f \in \MM$, where $\MM$ stands for some function space. Let $F \sqbrk f : \MM \to \R$ be a real-valued functional. Suppose: :$\forall f \in \MM: F \sqbrk f > 0$ Then $F$ is a '''positive-definite functional'''.
Definition:Earthworm Sequence
The '''earthworm sequence''' is the integer sequence $\sequence {h_n}$ defined as follows. Let $h: \Z \to \Z$ be the mapping defined as: :$\forall a \in \Z: \map h a = 2 a \pmod {100}$ Then: :$h_n = \begin {cases} x: x \in \set {10, 11, \dotsc, 99} & : n = 0 \\ \map h {n - 1} & : n > 0 \end {cases}$ That is: :$h_0$ is an arbitrary integer between $10$ and $99$ :$h_n$ is the result of multiplying the previous term in the sequence by $2$ and keeping only the last two digits.
Definition:Bounded Sequence/Normed Vector Space
Let $\struct {X, \norm {\, \cdot \,} }$ be a normed vector space. Let $\sequence {x_n}$ be a sequence in $X$. Then $\sequence {x_n}$ is '''bounded''' : :$\exists K \in \R$ such that $\forall n \in \N: \norm {x_n} \le K$
Definition:Bounded Sequence/Normed Vector Space/Unbounded
$\sequence {x_n}$ is '''unbounded'''  it is not bounded.
Definition:Quality of Integer Triple
Let $\tuple {a, b, c}$ be an ordered triple of (strictly) positive integers. The '''quality''' of $\tuple {a, b, c}$ is defined and denoted as: :$\map q {a, b, c} := \dfrac {\ln c} {\map \ln {\map {\operatorname {rad} } {abc} } }$ where $\operatorname {rad}$ denotes the radical of an integer.
Definition:Continuous Function Space
Let $X, Y$ be topological spaces. Let $f : X \to Y$ be a continuous mapping. Then the set of all such mappings $f$ is known as '''continuous function space''' and is denoted by $\CC \paren {X,Y}$: :$\CC \paren {X, Y} = \set {f : X \to Y}$
Definition:Bourbaki Dangerous Bend Symbol
The '''Bourbaki dangerous bend symbol''' is a symbol invented by the {{AuthorRef|Nicolas Bourbaki|Bourbaki group}} to warn the reader that there exists the serious possibility of making a common mistake while interpreting or utilising the material presented. It was originally presented by {{AuthorRef|Nicolas Bourbaki|Bourbaki}}, as the pre-$1949$ French traffic road sign: :20px Variants exist, notably {{AuthorRef|Donald Ervin Knuth}}'s version: :40px {{NamedforDef|Nicolas Bourbaki|cat = Bourbaki}}
Definition:Continuous Real-Valued Function Space
Let $X$ be a topological space. Let $f : X \to \R$ be a continuous real valued mapping. Then the set of all such mappings $f$ is known as '''continuous real-valued function space''' and is denoted by $\map \CC X$: :$\map \CC X := \map \CC {X, \R} = \set {f : X \to \R}$
Definition:Space of Continuous on Closed Interval Real-Valued Functions
Let $f : \closedint a b \to \R$ be a continuous real valued function. Then the set of all such mappings $f$ is known as '''continuous on closed interval real-valued function space''' and is denoted by $\CC \closedint a b$: :$\CC \closedint a b := \CC \paren {\closedint a b, \R} = \set {f : \closedint a b \to \R}$
Definition:Space of Continuous Functions of Differentiability Class k
Let $X, Y$ be normed vector spaces. Let $f : X \to Y$ be a mapping of a differentiability class $k \in \N_{>0}$ in the sense of Frechet. Then the set of all such mappings $f$ is known as '''continuous function space of differentiability class k''' and is denoted by $\CC^k \paren {X,Y}$: :$\CC^k \paren {X, Y} = \set {f : X \to Y}$
Definition:Real-Valued Periodic Function Space
Let $I = \closedint 0 T$ be a closed real interval. Let $\map \CC I$ be the space of real-valued functions, continuous on  $I$. Let the elements of $\map \CC I$ be real periodic functions with the period $T$:  :$f \in \map \CC I: \map f T = \map f 0$ Then the set of all such mappings $f$ is known as '''real-valued periodic (over $I$) function space''' and is denoted by $\map {\CC_{per}} I$: :$\map {\CC_{per}} I:= \CC_{per} \paren {I, \R} = \set {f : I \to \R : \map f T = \map f 0}$
Definition:Space of Real Polynomials of Degree n
Let $\map \CC \R$ be the space of real-valued continuous functions. Let $f \in \map \CC \R$ be a polynomial over real numbers of degree not greater than $n$. Then the set of all such mappings $f$ is known as '''space of real polynomials of degree $\le n$''' and is denoted by $\PP_n \paren {\R}$: :$\PP_n \paren {\R} := \set {f : \R \to \R : \map \deg f \le n}$
Definition:Space of Real Polynomials
Let $\PP_n$ be the space of real polynomials of degree not greater than n. Then the union of all such spaces is called the '''space of real polynomials''' and is denoted by $\PP$: :$\displaystyle \PP := \bigcup _{n \mathop = 0}^{\infty}\PP_n$
Definition:Jaina Mathematics
'''Jainism''' is a religion and philosophy which was founded in India around the $6$th century BCE. '''Jaina mathematics''' is mathematics done by those following Jainism. This has been practised on the Indian subcontinent from the founding of Jainism up to modern times.
Definition:Complete Lattice/Definition 1
Let $\struct {S, \preceq}$ be a lattice. Then $\struct {S, \preceq}$ is a '''complete lattice''' : :$\forall T \subseteq S: T$ admits both a supremum and an infimum.
Definition:Completely Hausdorff Space/Definition 1
$\struct {S, \tau}$ is a '''completely Hausdorff space''' or '''$T_{2 \frac 1 2}$ space''' : :$\forall x, y \in S, x \ne y: \exists U, V \in \tau: x \in U, y \in V: U^- \cap V^- = \O$  That is, for any two distinct elements $x, y \in S$ there exist open sets $U, V \in \tau$ containing $x$ and $y$ respectively whose closures are disjoint.
Definition:Completely Hausdorff Space/Definition 2
$\struct {S, \tau}$ is a '''completely Hausdorff space''' or '''$T_{2 \frac 1 2}$ space''' : :$\forall x, y \in S, x \ne y : \exists N_x, N_y \subseteq S: \exists U, V \in \tau: x \subseteq U \subseteq N_x, y \subseteq V \subseteq N_y: N_x^- \cap N_y^- = \O$ That is: :$\struct {S, \tau}$ is a '''$T_{2 \frac 1 2}$ space'''  every two points in $S$ are separated by closed neighborhoods.
Definition:Polyomino/Free
A '''free polyomino''' is a polyomino which is not distinguished from its image under a reflection. That is, it is considered '''free''' of the plane in which it is embedded, and can be "lifted up and turned over".
Definition:Ordering on Natural Numbers/Von Neumann Construction
Let $\omega$ denote the set of natural numbers as defined by the von Neumann construction. The '''strict ordering''' of $\omega$ is the relation $<$ defined by: :$\forall m, n \in \omega: m < n \iff m \subsetneq n$ The '''(weak) ordering''' of $\omega$ is the relation $\le$ defined by: :$\forall m, n \in \omega: m \le n \iff m \subseteq n$
Definition:Cardinality of Finite Class/Definition 1
Let $A$ be such that: :there exists a bijection $\phi$ from $A$ to $n$ where $n$ is a natural number as defined by the von Neumann construction. Then $A$ has '''cardinality $n$'''.
Definition:Cardinality of Finite Class/Definition 2
Let $A$ be such that: :there exists a bijection $\phi$ from $A$ to the set $\set {1, 2, \dotsc, n} = n^+ \setminus \set 0$ where: :$n$ is a natural number as defined by the von Neumann construction :$n^+$ is the successor of $n$. Then $A$ has '''cardinality $n$'''.
Definition:Finite Class
Let $A$ be a class. Let there exist a natural number $n$ as defined by the von Neumann construction such that there exists a bijection $\phi$ from $A$ to $n$. Then $A$ is a '''finite class'''.
Definition:Infinite Class
Let $A$ be a class. $A$ is an '''infinite class'''  $A$ is not a finite class.
Definition:Statistic
A '''statistic''' is broadly defined as '''a quantity which is calculated from sample data.
Definition:Dispersion (Statistics)
Let $S$ be a sample of a population in the context of statistics. The '''dispersion''' of $S$ is a general term meaning how much the data describing the sample are spread out. The word can also be applied to a random variable.
Definition:Denumerable Class
Let $A$ be a class. $A$ is '''denumerable'''  there exists a bijection: :$f: S \to \N$ where $\N$ is the set of natural numbers.
Definition:Non-Denumerable Class
A class which is neither denumerable nor finite is described as '''non-denumerable'''.
Definition:Machine (Mechanics)
A '''machine''', in the context of mechanics, is a physical artefact which takes energy in a particular form and converts it into energy in another form for a specific purpose.
Definition:Mach Number
'''Mach number''' is a unit of measurement of speed of a body $B$ in a fluid $F$. It is defined as the ratio of the speed of $B$ to the speed of sound in $F$. {{NamedforDef|Ernst Waldfried Josef Wenzel Mach|cat = Mach}}
Definition:Supersonic
The speed of a body $B$ is defined as '''supersonic'''  its Mach number is greater than $1$. That is, if $B$ is moving faster than the speed of sound in the fluid in which it is moving.
Definition:Rest
=== At Rest === The term '''at rest''' is a synonym for '''stationary''': {{:Definition:At Rest}} === Rest Mass === {{:Definition:Rest Mass}}
Definition:Electric Charge/Dimension
The dimension of measurement of '''electric charge''' is $\mathsf {I T}$.
Definition:Electric Charge/Units
The SI unit of '''electric charge''' is the coulomb $\mathrm C$.
Definition:Electric Field Strength/Units
The SI unit for '''electric field strength''' can be given either as: :volt per metre: $\mathrm V / \mathrm m$ :newton per coulomb: $\mathrm N / \mathrm C$
Definition:Electric Field Strength/Dimension
'''Electric field strength''' has the dimension $\mathsf {M L T}^{-3} \mathsf I^{-1}$.
Definition:Electric Field Strength
'''Electric field strength''' is the measure of the intensity of an electric field.
Definition:Electrostatics
'''Electrostatics''' is the branch of physics concerned with the study of stationary electrically charged bodies, and the forces and fields they give rise to.
Definition:Electric Charge/Sign
The '''sign''' of an '''electric charge''' can be one of $2$ types: === Positive Electric Charge === {{:Definition:Electric Charge/Sign/Positive}} === Negative Electric Charge === {{:Definition:Electric Charge/Sign/Negative}}
Definition:Electric Charge/Sign/Like
If $2$ electric charges are of the same sign, they are referred to as being '''like (electric) charges'''.
Definition:Electric Charge/Sign/Unlike
If $2$ electric charges are of opposite sign, they are referred to as being '''unlike (electric) charges'''.
Definition:Electric Charge/Sign/Positive
A '''positive electric charge''' is an electric charge which is of the same sign as the electric charge on a proton. When it is necessary to assign a value to a '''positive electric charge''', a $+$ (plus) sign is used, and the value assigned is a positive number.
Definition:Electric Charge/Sign/Negative
A '''negative electric charge''' is an electric charge which is of the same sign as the electric charge on an electron. When it is necessary to assign a value to a '''negative electric charge''', a $-$ (minus) sign is used, and the value assigned is a negative number.
Definition:Electric Charge/Sign/Convention
It appears at first glance that the definition of the sign of an electric charge is both arbitrary and circular. That is: :a positive electric charge is the type of electric charge that can be found on a proton while: :a proton is defined as carrying a positive electric charge. Early in the development of the theory of electrostatics, it was noticed that there were two types of electric charge. What they were called was at that point arbitrary; one type was called '''positive''', and the other type was called '''negative'''. When the proton and electron were discovered, the signs of their electric charges were identified, and subsequently used to ''define'' the sign of an arbitrary electric charge. Hence the fact that a proton is positive and an electron is negative is simply the result of a convention.
Definition:Coulomb (Unit)/Base Units
The SI base units of the '''coulomb''' are: :$\mathrm C := \mathrm A \, \mathrm s$ where: :$\mathrm A$ denotes amperes :$\mathrm s$ denotes seconds (of time).
Definition:Bounded Mapping/Real-Valued/Definition 3
$f$ is '''bounded on $S$''' : :$\exists a, b \in \R_{\ge 0}: \forall x \in S: \map f x \in \closedint a b$ where $\closedint a b$ denotes the (closed) real interval from $a$ to $b$.
Definition:Negation Function
The '''negation function''' is the function defined on the various standard number systems as follows: === Integer Negation Function === {{:Definition:Negation Function/Integer}} === Rational Negation Function === {{:Definition:Negation Function/Rational}} === Real Negation Function === {{:Definition:Negation Function/Real}} === Complex Negation Function === {{:Definition:Negation Function/Complex}}
Definition:Negation Function/Integer
The negation function $h: \Z \to \Z$ is defined on the set of integers as: :$\forall n \in \Z: \map h n = -n$
Definition:Union Mapping/Finite Set
Let $S = \set {f_1, f_2, \ldots, f_n}$ denote a finite set of mappings. The '''union mapping''' $f$ of $S$ is defined when: :$\forall i, j \in \set {1, 2, \ldots, n}: f_i$ and $f_j$ are combinable and is defined as: :$\forall x \in \displaystyle \bigcup \set {\Dom {f_i}: i \in \set {1, 2, \ldots, n} } x \in \Dom {f_i} \implies f = \map {f_i} x$
Definition:Union Mapping/Family
Let $I$ be an indexing set. Let $F = \family {f_i}_{i \mathop \in I}$ be a family of mappings indexed by $I$ The '''union mapping''' $f$ of $F$ is defined when: :$\forall i, j \in I: f_i$ and $f_j$ are combinable and is defined as: :$\forall x \in \displaystyle \bigcup \set {\Dom {f_i}: i \in I} x \in \Dom {f_i} \implies f = \map {f_i} x$
Definition:Set Product/Projection
The mappings $\phi_i$ are the '''projections''' of $P$.
Definition:Group Product/Group Law
The operation $\circ$ can be referred to as the '''group law'''.
Definition:Group Product/Product Element
Let $a, b \in G$ such that $ = a \circ b$. Then $g$ is known as the '''product of $a$ and $b$'''.
Definition:Closed under Inversion
Let $\struct {G, \circ}$ be an group with an identity element $e$. Let $H \subseteq G$ be a subset of $G$. Then $H$ is '''closed under inversion''' : :$\forall h \in H: h^{-1} \in H$ That is,  the inverse of every element of $H$ is itself in $H$.
Definition:Torus Group
The '''torus group''' is the direct product of the circle group $C$ with itself: :$T := \struct {K, \times} \times \struct {K, \times}$ where: :$K = \set {z \in \C: \cmod z = 1}$
Definition:Closed Set/Normed Vector Space/Definition 1
'''$F$ is closed in $V$'''  its complement $X \setminus F$ is open in $V$.
Definition:Closed Set/Normed Vector Space/Definition 2
'''$F$ is closed (in $V$)'''  every limit point of $F$ is also a point of $F$. That is:  $F$ contains all its limit points.
Definition:Simple Graph/Formal Definition
Let $V$ be a set. Let $\RR$ be an endorelation on $V$ which is antireflexive and symmetric. Let $E$ be the set whose elements of the form: :$\set {\tuple {v_a, v_b}, \tuple {v_b, v_a} }$. where $\tuple {v_a, v_b}$ and $\tuple {v_b, v_a}$ are elements of $\RR$  A '''simple graph''' is an ordered pair $G = \struct {V, E}$, where $V$ and $E$ are defined as above. $V$ is called the vertex set. $E$ is called the edge set.
Definition:Directed Graph/Formal Definition
A '''directed graph''' or '''digraph''' $D$ is a non-empty set $V$ together with an antireflexive relation $E$ on $V$. The elements of $E$ are the arcs.
Definition:Directed Graph/Simple Digraph
If the relation $E$ in $D$ is also specifically asymmetric, then $D$ is called a '''simple digraph'''. That is, in a '''simple digraph''' there are no pairs of arcs (like there are between $v_1$ and $v_4$ in the diagram above) which go in both directions between two vertices.
Definition:Star Graph
The '''star graph of order $n$''', denoted $S_n$ is a simple graph with $n$ vertices with the following properties: :One distinguished vertex is of degree $n - 1$ :The remaining vertices are all of degree $1$ and are adjacent only to the distinguished vertex. :700px
Definition:Loop-Graph/Formal Definition
A '''loop-graph''' $G$ is a non-empty set $V$ together with a symmetric relation $E$ on $G$. Thus it can be seen that a loop-graph is a simple graph with the stipulation that the relation $E$ does not need to be antireflexive.
Definition:Bounded Normed Vector Space
Let $M = \struct {X, \norm {\, \cdot \,}}$ be a normed vector space. Let $M' = \struct {Y, \norm {\, \cdot \,}_Y}$ be a normed vector subspace of $M$. === Definition 1 === {{:Definition:Bounded Normed Vector Space/Definition 1}} === Definition 2 === {{:Definition:Bounded Normed Vector Space/Definition 2}}
Definition:Bounded Normed Vector Space/Definition 1
'''$M'$ is bounded (in $M$)''' : :$\exists x \in X, C \in \R: \forall y \in Y: \norm {x - y} \le C$
Definition:Bounded Normed Vector Space/Definition 2
'''$M'$ is bounded (in $M$)''' : :$\exists \epsilon \in \R_{>0} : \exists x \in X : Y \subseteq \map {B_\epsilon^-} x$ where $\map {B_\epsilon^-} x$ is a closed ball in $M$.
Definition:Loop-Graph/Loop-Digraph/Formal Definition
A '''loop-digraph''' $D$ is a non-empty set $V$ together with a relation $E$ on $D$. Thus it can be seen that a '''loop-digraph''' is a directed graph with the stipulation that the relation $E$ does not need to be antireflexive.
Definition:Loop-Graph/Loop-Digraph/Loop
Such an arc is called a loop.
Definition:Radiometric Dating
'''Radiometric dating''' is a technique whose purpose is to work out the age $T$ of a physical object $B$. The pieces of information are known: :$(1): \quad$ The ratio $r_0$ of a radioactive isotope $E_R$ of a chemical element $E$ to its stable isotope $E_S$ in $B$ at the time it came into being :$(2): \quad$ The ratio $r$ of $E_R$ to $E_S$ at the time now :$(3): \quad$ The half-life of the radioactive isotope $E_R$ It is known from scientific investigation what $r_0$ is when a physical object is created. It is also known from scientific investigation what the rate of radioactive decay of $E_R$ is. Hence it can be worked out by use of the First-Order Reaction how long it would take for the ratio of $E_R$ to $E_S$ to reach its current ratio $r$.
Definition:Radiometric Dating/Radiocarbon Dating
'''Radiocarbon dating''' is a specific application of '''radiometric dating''' which is used to determine how long a piece of organic matter has been dead. Because of bombardment by cosmic rays, the ratio of (radioactive) carbon-14 to (stable) carbon-12 in the atmosphere of Earth is fairly constant. This ratio is known. The ratio of carbon-14 to carbon-12 in a living organism is the same as it is in Earth's atmosphere, by biological respiration. However, when the organism dies, it no longer respires, and the carbon in its body stays where it was at the time of its death. As time passes, the carbon-14 decays to nitrogen-14 via the process of beta decay. The half-life of carbon-14 is known to be $5700 \pm 40$ years.
Definition:P-adic Number/Quotient of Cauchy Sequences in P-adic Norm/Representative
Each Cauchy sequence $\sequence {y_n}$ of the left coset $\sequence {x_n} + \NN$ is called a '''representative''' of the $p$-adic number $\sequence {x_n} + \NN$.
Definition:Real Function/Definition 1
Let $S \subseteq \R$ be a subset of the set of real numbers $\R$. Suppose that, for each value of the independent variable $x$ of $S$, there exists a corresponding value of the dependent variable $y$. Then the dependent variable $y$ is a '''(real) function''' the independent variable $x$.
Definition:Real Function/Range
Let $f: S \to \R$ be a real function. The range of $f$ is the set of values that the dependent variable can take.
Definition:Real Function/Definition 2
A '''(real) function''' is correspondence between a domain set $D$ and a set range set $R$ that assigns to each element of $D$ a unique element of $R$.
Definition:Implicit Function/General
Let: :$f: \R^{n + 1} \to \R, \tuple {x_1, x_2, \ldots, x_n, z} \mapsto \map f {x_1, x_2, \ldots, x_n, z}$ be a real-valued function on $\R^{n + 1}$, where: :$\tuple {x_1, x_2, \ldots, x_n} \in \R^n, z \in \R$ Let a relation between $x_1, x_2, \ldots, x_n$ and $z$ be expressed in the form: :$\map f {x_1, x_2, \ldots, x_n, z} = 0$ defined on some subset $S \subseteq \R^n$. If there exists a function $g: S \to \R$ such that: :$\forall \tuple {x_1, x_2, \ldots, x_n} \in S: z = \map g {x_1, x_2, \ldots, x_n} \iff \map f {x_1, x_2, \ldots, x_n, z} = 0$ then the relation $\map f {x_1, x_2, \ldots, x_n, z} = 0$ defines $z$ as an '''implicitly defined function''' of $x_1, x_2, \ldots, x_n$.
Definition:Real Function/Two Variables/Substitution for y
Let $f: S \times T \to \R$ be a (real) function of two variables: :$z = \map f {x, y}$ Then: :$\map f {x, a}$ means the real function of $x$ obtained by replacing the independent variable $y$ with $a$. In this context, $a$ can be: :a real constant such that $a \in T$ :a real function $\map g x$ whose range is a subset of $T$.
Definition:Bounded Region of Plane
Let $D \subseteq \R^2$ be a subset of the plane. $D$ is '''bounded'''  there exists a circle in the plane which completely encloses $D$.