hcju/proofwikips huggingface dataset:
Definition:Stratified Random Sample/Strata
The components of $P$ are called '''strata''', singular '''stratum'''.
Definition:Dual Statement (Category Theory)
=== Morphisms-Only Category Theory === Let $\Sigma$ be a statement in the language of category theory. The '''dual statement''' $\Sigma^*$ of $\Sigma$ is the statement obtained from substituting: {{begin-eqn}} {{eqn|l = R_\circ \left({y, x, z}\right)      |o = \) for \(      |r = R_\circ \left({x, y, z}\right) }} {{eqn|l = \operatorname{dom}      |o = \) for \(      |r = \operatorname{cod} }} {{eqn|l = \operatorname{cod}      |o = \) for \(      |r = \operatorname{dom} }} {{end-eqn}} === Object Category Theory === In the more convenient description of metacategories by using objects, the '''dual statement''' $\Sigma^*$ of $\Sigma$ then becomes the statement obtained from substituting: {{begin-eqn}} {{eqn|l = f \circ g      |o = \) for \(      |r = g \circ f }} {{eqn|l = \operatorname{cod}      |o = \) for \(      |r = \operatorname{dom} }} {{eqn|l = \operatorname{dom}      |o = \) for \(      |r = \operatorname{cod} }} {{end-eqn}} {{finish|a page proving that these indeed are the resulting substitutions}} === Example === For example, if $\Sigma$ is the statement: :$\exists g: g \circ f = \operatorname{id}_{\operatorname{dom} f}$ describing that $f$ is a split mono, then $\Sigma^*$ becomes: :$\exists g: f \circ g = \operatorname{id}_{\operatorname{cod} f}$ which precisely expresses $f$ to be a split epi. For a set $\mathcal E$ of statements, write: :$\mathcal E^* := \left\{{\Sigma^*: \Sigma \in \mathcal E}\right\}$  for the set comprising of the dual statement of those in $\mathcal E$.
Definition:Cartesian Product/Factors
Let $S \times T$ be the cartesian product of $S$ and $T$. Then the sets $S$ and $T$ are called the '''factors''' of $S \times T$.
Definition:Metamodel
Let $\mathcal L$ be a formal system with deductive apparatus $\mathcal D$. A '''metamodel''' $\mathcal M$ for $\mathcal L$ is an interpretation for $\mathcal L$. Thus, it assigns meanings to all of the undefined terms of $\mathcal L$. Further, it specifies a means for deciding if a sentence of $\mathcal L$ is to be considered true in $\mathcal M$. It is then required that all the sentences derivable by the deductive apparatus $\mathcal D$ are true in $\mathcal M$. From the above specification, it follows that a '''metamodel''' $\mathcal M$ is a conceptual device, in that it does not require to be founded on rigorous mathematical foundations. Thus, $\mathcal M$ need not be a set or some other rigid notion, just something that one can contemplate, denote and convey to others. For example, even without rigid mathematical foundations it is clear to all sane people that there is a concept "set" that one can reason about. One could then try to determine if the collection of these "sets" is a '''metamodel''' for the language of set theory. As another example there is the notion of metacategory, which can be described as being any '''metamodel''' for the language of category theory.
Definition:QED
The initials of '''Quod Erat Demonstrandum''', which is {{WP|Latin|Latin}} for '''which was to be demonstrated'''. These initials were traditionally added to the end of a proof, after the last line which is supposed to contain the statement that was to be proved.
Definition:Real Interval/Unit Interval
A '''unit interval''' is a real interval whose endpoints are $0$ and $1$: {{begin-eqn}} {{eqn | l = \openint 0 1       | o = :=       | r = \set {x \in \R: 0 < x < 1} }} {{eqn | l = \hointr 0 1       | o = :=       | r = \set {x \in \R: 0 \le x < 1} }} {{eqn | l = \hointl 0 1       | o = :=       | r = \set {x \in \R: 0 < x \le 1} }} {{eqn | l = \closedint 0 1       | o = :=       | r = \set {x \in \R: 0 \le x \le 1} }} {{end-eqn}}
Definition:Inductive Set/Subset of Real Numbers
Let $I$ be a subset of the real numbers $\R$. Then $I$ is an '''inductive set''' : :$1 \in I$  and :$x \in I \implies \paren {x + 1} \in I$
Definition:Language of Category Theory
The language of '''(morphisms-only) category theory''' is a specific instance of the language of predicate logic. As such, it is considered to have: * unary function symbols $\operatorname{dom}$ and $\operatorname{cod}$, called the '''domain''' and '''codomain''' symbols * A ternary relation symbol $R_\circ$, called the '''composition relation''' symbol All further descriptions required to determine a formal language are inherited from the definition of the language of predicate logic.
Definition:Morphisms-Only Metacategory
A '''morphisms-only metacategory''' is a metamodel for the language of category theory subject to the following axioms: {{:Axiom:Axioms for Morphisms-Only Category Theory}}
Definition:Mapping/Defined
A mapping $f \subseteq S \times T$ is '''defined''' at $x \in S$ : :$\exists y \in T: \tuple {x, y} \in f$ If for some $x \in S$, one has: :$\forall y \in T: \tuple {x, y} \notin f$ then $f$ is '''not defined''' or ('''undefined''') at $x$, and indeed, $f$ is not technically a mapping at all.
Definition:Relation/Truth Set
Let $\RR$ be a relation on $S \times T$. The '''truth set''' of $\RR$ is the set of all ordered pairs $\tuple {s, t}$ of $S \times T$ such that $s \mathrel \RR t$: : $\map \TT \RR = \set {\tuple {s, t}: s \mathrel \RR t}$
Definition:Relation/General Definition
Let $\displaystyle \Bbb S = \prod_{i \mathop = 1}^n S_i = S_1 \times S_2 \times \ldots \times S_n$ be the cartesian product of $n$ sets $S_1, S_2, \ldots, S_n$. An '''$n$-ary relation on $\Bbb S$''' is an ordered $n + 1$-tuple $\RR$ defined as: :$\RR := \struct {S_1, S_2, \ldots, S_n, R}$ where $R$ is an arbitrary subset $R \subseteq \Bbb S$. To indicate that $\tuple {s_1, s_2, \ldots, s_n} \in R$, we write: :$\map \RR {s_1, s_2, \ldots, s_n}$
Definition:Relation/Unary Relation
As a special case of an $n$-ary relation on $S$, note that when $n = 1$ we define a '''unary relation''' on $S$ as: :$\RR \subseteq S$ That is, a '''unary relation''' is a subset of $S$.
Definition:Random Selection
A manner of selecting objects from some larger collection of objects is said to be '''random''' if the selection is made according to chance. That is, there is no strict rule or procedure that predictably determines the outcome of the selection.  See experiment and random variable for a  precise mathematical treatment of randomness.
Definition:Agreement/Relations
Let: :$(1): \quad \mathcal R_1 \subseteq S_1 \times T_1$ be a relation on $S_1 \times T_1$ :$(2): \quad \mathcal R_2 \subseteq S_2 \times T_2$ be a relation on $S_2 \times T_2$ :$(3): \quad X \subseteq S_1 \cap S_2$ Let: :$\forall s \in X: \mathcal R_1 \left ({s}\right) = \mathcal R_2 \left ({s}\right)$ where $\mathcal R_1 \left ({s}\right)$ denotes the image of $s$ under $\mathcal R$: :$\mathcal R_1 \left ({s}\right) := \left\{ {t \in T: s \mathrel{\mathcal R_1} t}\right\}$ Then the relations $\mathcal R_1$ and $\mathcal R_2$ are said to '''agree on''' or '''be in agreement on''' $X$.
Definition:Agreement/Mappings
Let: :$(1): \quad f_1: S_1 \to T_1$ be a mapping from $S_1$ to $T_1$ :$(2): \quad f_2: S_2 \to T_2$ be a mapping from $S_2$ to $T_2$ :$(3): \quad X \subseteq S_1 \cap S_2$ Let: :$\forall s \in X: \map {f_1} s = \map {f_2} s$ Then the mappings $f_1$ and $f_2$ are said to '''agree on''' or '''be in agreement on''' $X$.
Definition:Combinable/Mappings
Let: :$(1): \quad f_1: S_1 \to T_1$ be a mapping from $S_1$ to $T_1$ :$(2): \quad f_2: S_2 \to T_2$ be a mapping from $S_2$ to $T_2$ If $f_1$ and $f_2$ agree on $S_1 \cap S_2$, they are said to be '''combinable'''.
Definition:Dedekind Complete Set
Let $\struct {S, \preceq}$ be an ordered set. Then $\struct {S, \preceq}$ is '''Dedekind complete'''  every non-empty subset of $S$ that is bounded above admits a supremum (in $S$).
Definition:Dedekind-MacNeille Completion
Let $\struct {S, \preceq}$ be an ordered set. For a subset $A \subseteq S$, let $A_+$ and $A_-$ be the sets of all upper and lower bounds for $A$ in $S$, respectively. {{explain|Review notation for upper and lower bounds}} The '''Dedekind-MacNeille completion''' of $\struct {S, \preceq}$ is defined as the set: :$\widehat S := \set {A \subseteq S: A = \paren {A_+}_-}$ ordered by inclusion ($\subseteq$). {{explain|ordered by inclusion in this context - exactly how does this work?}}
Definition:Dedekind Completion
Let $S$ be an ordered set. A '''Dedekind completion of $S$''' is a Dedekind complete ordered set $\tilde S$ together with an order embedding $\phi: S \to \tilde S$, subject to: :For all Dedekind complete ordered sets $X$, and for all order embeddings $f: S \to X$, there exists a unique order embedding $\tilde f: \tilde S \to X$ such that: ::$\tilde f \circ \phi = f$
Definition:Abnormal Subgroup
Let $G$ be a group. Let $H$ be a subgroup of $G$. Then $H$ is an '''abnormal subgroup''' : :$\forall g \in G: g \in \gen {H, H^g}$ where $\gen {H, H^g}$ is the subgroup of $G$ generated by $H$ and the conjugate of $H$ by $g$.
Definition:Self-Normalizing Subgroup
Let $G$ be a group. Let $H$ be a subgroup of $G$. Then $H$ is a '''self-normalizing subgroup''' : :$N_G \left({H}\right) = H$ where $N_G \left({H}\right)$ is the normalizer of $H$ in $G$.
Definition:Contranormal Subgroup
Let $G$ be a group. Let $H$ be a subgroup of $G$. Then $H$ is a '''contranormal subgroup'''  the normal closure of $H$ is the whole of $G$.
Definition:Subnormal Subgroup
Let $G$ be a group. Let $H$ be a subgroup of $G$. Then $H$ is a '''$k$-subnormal subgroup of $G$'''  there is a finite sequence of subgroups of $G$: :$H = H_0, H_1, H_2, \ldots, H_k = G$ such that $H_i$ is a normal subgroup of $H_{i+1}$ for all $i \in \left\{{0, 1, \ldots, k-1}\right\}$. That is,  there exists a normal series: :$H = H_0 \lhd H_1 \lhd H_2 \lhd \cdots \lhd H_k = G$ where $\lhd$ denotes the relation of normality. Thus $H$ is a '''subnormal subgroup of $G$'''  there exists $k \in \Z_{>0}$ such that $H$ is a '''$k$-subnormal subgroup of $G$'''.
Definition:Pronormal Subgroup
Let $G$ be a group. Let $H$ be a subgroup of $G$. Then $H$ is a '''pronormal subgroup in $G$''' iff each of its conjugates in $G$ is conjugate to it already in the subgroup generated by $H$ and its conjugate. That is, $H$ is '''pronormal in $G$''' iff: : $\forall g \in G: \exists k \in \left\langle{H, H^g}\right\rangle: H^k = H^g$ where: : $\left\langle{H, H^g}\right\rangle$ is the subgroup generated by $H$ and $H^g$ : $H^g$ is the conjugate of $H$ by $g$.
Definition:Paranormal Subgroup
Let $G$ be a group. Let $H$ be a subgroup of $G$. Then $H$ is a '''paranormal subgroup in $G$'''  the subgroup generated by $H$ and any conjugate of $H$ is also generated by $H$ and a conjugate of $H$ within that generated subgroup. That is, $H$ is '''paranormal in $G$''' : : $\forall g \in G: \exists k \in \left\langle{H, H^g}\right\rangle: \left\langle{H, H^k}\right\rangle = \left\langle{H, H^g}\right\rangle$ where: : $\left\langle{H, H^g}\right\rangle$ is the subgroup generated by $H$ and $H^g$ : $H^g$ is the conjugate of $H$ by $g$.
Definition:Polynormal Subgroup
Let $G$ be a group. Let $H$ be a subgroup of $G$. Then $H$ is a '''polynormal subgroup''' of $G$  for all $g \in G$, the conjugate closure of $H$ in $\langle H, g \rangle$ is equal to the conjugate closure of $H$ in $H^{\langle g \rangle}$. That is, $H$ is '''polynormal in $G$''' : : $\forall g \in G: H^{\left\langle g \right\rangle} = H^{H^{\left\langle g \right\rangle}}$ where: : $H^{\left\langle g \right\rangle}$ is the subgroup generated by the set of all elements of the form $g^nhg^{-n}$ where $h \in H, n \in \mathbb{Z}$, : $H^{H^{\left\langle g \right\rangle}}$ is the subgroup generated by the set of all elements of the form $khk^{-1}$ where $k \in H^{\left\langle g \right\rangle}$.
Definition:Underlying Set/Abstract Algebra
Let $\struct {S, \circ}$ be an algebraic structure. Then the '''underlying set''' of $\struct {S, \circ}$ is the set $S$.
Definition:Underlying Set/Metric Space
Let $M = \left({A, d}\right)$ be a metric space. Then the '''underlying set''' of $M$ is the set $A$.
Definition:Underlying Set/Topological Space
Let $T = \left({S, \tau}\right)$ be a topological space. Then the '''underlying set''' of $T$ is the set $S$.
Definition:Multiplication/Complex Numbers
The '''multiplication operation''' in the domain of complex numbers $\C$ is written $\times$. Let $z = a + i b, w = c + i d$ where $a, b, c, d \in \R$. Then $z \times w$ is defined as: :$\paren {a + i b} \times \paren {c + i d} = \paren {a c - b d} + i \paren {a d + b c}$
Definition:Coproduct
Let $\mathbf C$ be a metacategory. Let $A$ and $B$ be objects of $\mathbf C$. A '''coproduct diagram''' for $A$ and $B$ comprises an object $P$ and morphisms $i_1: A \to P$, $i_2: B \to P$: ::$\begin{xy} <-4em,0em>*+{A} = "A", <0em,0em>*+{P}  = "P", <4em,0em>*+{B}  = "B", "A";"P" **@{-} ?>*@{>} ?*!/^.8em/{i_1}, "B";"P" **@{-} ?>*@{>} ?*!/_.8em/{i_2}, \end{xy}$ subjected to the following universal mapping property: :For any object $X$ and morphisms $x_1, x_2$ like so: ::$\begin{xy} <-4em,0em>*+{A} = "A", <0em,0em>*+{X}  = "X", <4em,0em>*+{B}  = "B", "A";"X" **@{-} ?>*@{>} ?*!/^.8em/{x_1}, "B";"X" **@{-} ?>*@{>} ?*!/_.8em/{x_2}, \end{xy}$ :there is a unique morphism $u: P \to X$ such that: ::$\begin{xy} <0em,5em>*+{X}  = "X", <-5em,0em>*+{A} = "A", <0em,0em>*+{P}  = "P", <5em,0em>*+{B}  = "B", "A";"X" **@{-}  ?>*@{>} ?*!/^.8em/{x_1}, "B";"X" **@{-}  ?>*@{>} ?*!/_.8em/{x_2}, "P";"X" **@{--} ?>*@{>} ?*!/_.6em/{u}, "A";"P" **@{-}  ?>*@{>} ?*!/_.8em/{i_1}, "B";"P" **@{-}  ?>*@{>} ?*!/^.8em/{i_2}, \end{xy}$ :is a commutative diagram. That is: :$x_1 = u \circ i_i$ and $x_2 = u \circ i_2$ In this situation, $P$ is called a '''coproduct of $A$ and $B$''' and may be denoted $A + B$. We generally write $\sqbrk {x_1, x_2}$ for the unique morphism $u$ determined by above diagram. The morphisms $i_1$ and $i_2$ are often taken to be implicit. They are called '''injections'''; if necessary, $i_1$ can be called the '''first injection''' and $i_2$ the '''second injection'''. {{expand|the injection definition may merit its own, separate page}} {{definition wanted|general definition}} === Coproduct of Sets === When the objects $A$ and $B$ are sets, the definition of '''coproduct''' takes on the following form. {{:Definition:Coproduct/Sets}}
Definition:Variable
A '''variable''' is a label which is used to refer to an unspecified object. A '''variable''' can be identified by means of a symbol, for example: :$x, y, z, A, B, C, \phi, \psi, \aleph$ It is often convenient to append a subscript letter or number to distinguish between different objects of a similar type: :$a_0, a_1, a_2, \ldots, a_n; S_\phi, S_{\phi_x}, \ldots$ The type of symbol used to define a '''variable''' is purely conventional. Particular types of object, as they are introduced, frequently have a particular range of symbols specified to define them, but there are no strict rules on the subject.
Definition:Variable/Descriptive Statistics
A '''variable''' is a characteristic property of all individuals in a population or sample. It is a categorization of the population such that each individual can be unambiguously described with respect to said variable. === Quantitative Variable === {{:Definition:Variable/Descriptive Statistics/Quantitative Variable}} === Qualitative Variable === {{:Definition:Variable/Descriptive Statistics/Qualitative Variable|Qualitative Variable}}
Definition:Variable/Descriptive Statistics/Quantitative Variable
A '''quantitative variable''' is a variable such that: : The variable can be described by numbers : The performing of arithmetical operations on the data is meaningful.
Definition:Variable/Descriptive Statistics/Qualitative Variable
A '''qualitative variable''' is a variable such that: : The variable is not a quantitative variable : The variable describes each individual as either having, or not having, some specific property.
Definition:Basis (Topology)/Analytic Basis
{{:Definition:Basis (Topology)/Analytic Basis/Definition 1}}
Definition:Basis (Topology)/Synthetic Basis
Let $S$ be a set. ==== Definition 1 ==== {{:Definition:Basis (Topology)/Synthetic Basis/Definition 1}} ==== Definition 2 ==== {{:Definition:Basis (Topology)/Synthetic Basis/Definition 2}}
Definition:Countable Basis
A '''countable (analytic) basis''' for a topology $\tau$ is an analytic basis for $\tau$ which is a countable set.
Definition:Sub-Basis/Analytic Sub-Basis
Let $\struct {S, \tau}$ be a topological space. Let $\SS \subseteq \tau$. Define: :$\displaystyle \BB = \set {\bigcap \FF: \FF \subseteq \SS, \FF \text{ is finite} }$ That is, $\BB$ is the set of all finite intersections of sets in $\SS$. Note that $\FF$ is allowed to be empty in the above definition. Define: :$\displaystyle \tau' = \set {\bigcup \AA: \AA \subseteq \BB}$ Suppose that $\tau \subseteq \tau'$. That is, suppose that every $U \in \tau$ is a union of finite intersections of sets in $\SS$, together with $\O$ and $S$ itself. Then $\SS$ is called an '''analytic sub-basis''' for $\tau$.
Definition:Sub-Basis/Synthetic Sub-Basis
Let $X$ be a set. A '''synthetic sub-basis on $X$''' is ''any'' subset $\mathcal S \subseteq \mathcal P \left({X}\right)$ of the power set of $X$.
Definition:Differential/Real Function
=== At a Point === {{:Definition:Differential/Real Function/Point}} === On an Open Set === {{:Definition:Differential/Real Function/Open Set}} Also: :$f \left({x + h}\right) - f \left({x}\right) - \mathrm d f \left({x; h}\right) = o \left({h}\right)$ as $h \to 0$. In the above, $o \left({h}\right)$ is interpreted as little-O of $h$. {{refactor|put the above somewhere else}}
Definition:Differential/Real-Valued Function
{{:Definition:Differential/Real-Valued Function/Point}}
Definition:Differential/Vector-Valued Function
=== At a Point === {{:Definition:Differential/Vector-Valued Function/Point}} === On an Open Set === Let $O \subseteq \R^n$ be an open set. Let $f = \tuple {f_1, \ldots, f_m}^\intercal: O \to \R^m$ be a vector valued function, differentiable at $x \in O$. The '''differential''' $\d f$ is a function of two variables, defined as: :$\map {\d f} {x; h} = \map {J_f} x \cdot h$ where $\map {J_f} x$ be the Jacobian matrix of $f$ at $x$. That is, if $h = \tuple {h_1, \ldots, h_n}$: :$\map {\d f} {x; h} = \begin {pmatrix} \map {\dfrac {\partial f_1} {\partial x_1} } x & \cdots & \map {\dfrac {\partial f_1} {\partial x_n} } x \\ \vdots & \ddots & \vdots \\ \map {\dfrac {\partial f_m} {\partial x_1} } x & \cdots & \map {\dfrac {\partial f_m} {\partial x_n} } x \end {pmatrix} \begin {pmatrix} h_1 \\ \vdots \\ h_n \end {pmatrix}$
Definition:Dependent Variable/Real Function
Let $f: \R \to \R$ be a real function. Let $\map f x = y$. Then $y$ is referred to as a '''dependent variable'''.
Definition:Independent Variable/Real Function
Let $f: \R \to \R$ be a real function. Let $\map f x = y$. Then $x$ is referred to as an '''independent variable'''.
Definition:Jacobian
Let $U$ be an open subset of $\R^n$. Let $\mathbf f = \paren {f_1, f_2, \ldots, f_m}^\intercal: U \to \R^m$ be a vector valued function, differentiable at $\mathbf x = \paren {x_1, x_2, \ldots, x_n}^\intercal \in U$.
Definition:Jacobian/Matrix
The '''Jacobian matrix''' of $\mathbf f$ at $\mathbf x$ is defined to be the matrix of partial derivatives: :$\displaystyle \mathbf J_{\mathbf f} := \begin{pmatrix}  \map {\dfrac {\partial f_1} {\partial x_1} } {\mathbf x} & \cdots & \map {\dfrac {\partial f_1} {\partial x_n} } {\mathbf x} \\  \vdots & \ddots & \vdots \\  \map {\dfrac {\partial f_m} {\partial x_1} } {\mathbf x} & \cdots & \map {\dfrac {\partial f_m} {\partial x_n} } {\mathbf x} \end{pmatrix}$
Definition:Jacobian/Determinant
The '''Jacobian determinant''' of $\mathbf f$ at $\mathbf x$ is defined to be the determinant of the Jacobian matrix: :$\displaystyle \map \det {\mathbf J_{\mathbf f} } := \begin{vmatrix}  \map {\dfrac {\partial f_1} {\partial x_1} } {\mathbf x} & \cdots & \map {\dfrac {\partial f_1} {\partial x_n} } {\mathbf x} \\  \vdots & \ddots & \vdots \\  \map {\dfrac {\partial f_m} {\partial x_1} } {\mathbf x} & \cdots & \map {\dfrac {\partial f_m} {\partial x_n} } {\mathbf x} \end{vmatrix}$
Definition:Differentiable Mapping/Real Function
{{:Definition:Differentiable Mapping/Real Function/Point}}
Definition:Differentiable Mapping/Complex Function
{{:Definition:Differentiable Mapping/Complex Function/Point}}
Definition:Differentiable Mapping/Real-Valued Function
{{:Definition:Differentiable Mapping/Real-Valued Function/Point}}
Definition:Differentiable Mapping/Vector-Valued Function
{{:Definition:Differentiable Mapping/Vector-Valued Function/Point}}
Definition:Differentiable Mapping/Real Function/Point
Let $f$ be a real function defined on an open interval $\openint a b$. Let $\xi$ be a point in $\openint a b$. ==== Definition 1 ==== {{:Definition:Differentiable Mapping/Real Function/Point/Definition 1}} ==== Definition 2 ==== {{:Definition:Differentiable Mapping/Real Function/Point/Definition 2}} These limits, if they exist, are called the derivative of $f$ at $\xi$.
Definition:Differentiable Mapping/Real Function/Interval/Closed Interval
Let $f$ be differentiable on the open interval $\openint a b$. If the following limit from the right exists: :$\displaystyle \lim_{x \mathop \to a^+} \frac {\map f x - \map f a} {x - a}$ as well as this limit from the left: :$\displaystyle \lim_{x \mathop \to b^-} \frac {\map f x - \map f b} {x - b}$ then $f$ is '''differentiable on the closed interval $\closedint a b$'''.
Definition:Differentiable Mapping/Real Function/Interval
Let $f$ be a real function defined on an open interval $\openint a b$. Then $f$ is '''differentiable on $\openint a b$'''  $f$ is differentiable at each point of $\openint a b$.
Definition:Differentiable Mapping/Real Function/Real Number Line
Let $f$ be a real function defined on $\R$. By definition, $\R$ is an (unbounded) open interval. Let $f$ be differentiable on the open interval $\R$. That is, let $f$ be differentiable at every point of $\R$. Then $f$ is '''differentiable everywhere (on $\R$)'''.
Definition:Differentiable Mapping/Real-Valued Function/Point
Let $U$ be an open subset of $\R^n$.  Let $f: U \to \R$ be a real-valued function. Let $x \in U$. ==== Definition 1 ==== {{:Definition:Differentiable Mapping/Real-Valued Function/Point/Definition 1}} ==== Definition 2 ==== {{:Definition:Differentiable Mapping/Real-Valued Function/Point/Definition 2}}
Definition:Differentiable Mapping/Real-Valued Function/Open Set
Let $\mathbb X$ be an open subset of $\R^n$.  Let $f: \mathbb X \to \R$ be a real-valued function. Then $f$ is '''differentiable in the open set $\mathbb X$'''  $f$ is differentiable at each point of $\mathbb X$.
Definition:Differentiable Mapping/Vector-Valued Function/Point
Let $\mathbb X$ be an open subset of $\R^n$.  Let $f = \tuple {f_1, f_2, \ldots, f_m}^\intercal: \mathbb X \to \R^m$ be a vector valued function. ==== Definition 1 ==== {{:Definition:Differentiable Mapping/Vector-Valued Function/Point/Definition 1}} ==== Definition 2 ==== {{:Definition:Differentiable Mapping/Vector-Valued Function/Point/Definition 2}}
Definition:Differentiable Mapping/Vector-Valued Function/Region
Let $S \subseteq \mathbb X$. Then $f$ is '''differentiable in the open set $S$'''  $f$ is differentiable at each $x$ in $S$. This can be denoted $f \in \map {\CC^1} {S, \R^m}$.
Definition:Vector-Valued Function/Component Function
Each $f_1, f_2, \ldots, f_n$ is a '''component function''' of $\mathbf r$.
Definition:Subdivision (Real Analysis)/Normal Subdivision
$P$ is a '''normal subdivision of $\closedint a b$''' : :the length of every interval of the form $\closedint {x_i} {x_{i + 1} }$ is the same as every other. That is, : :$\exists c \in \R_{> 0}: \forall i \in \N_{< n}: x_{i + 1} - x_i = c$
Definition:Topology Generated by Synthetic Basis
Let $S$ be a set. Let $\BB$ be a synthetic basis of $S$. === Definition 1 === {{:Definition:Topology Generated by Synthetic Basis/Definition 1}} === Definition 2 === {{:Definition:Topology Generated by Synthetic Basis/Definition 2}} === Definition 3 === {{:Definition:Topology Generated by Synthetic Basis/Definition 3}}
Definition:Complex Conjugate/Complex Conjugation
The operation of '''complex conjugation''' is the mapping: : $\overline \cdot: \C \to \C: z \mapsto \overline z$. where $\overline z$ is the complex conjugate of $z$. That is, it maps a complex number to its complex conjugate.