nyuuzyou/wikis huggingface dataset:
Distance from Subset of Real Numbers to Element/Examples/From 3 to Open Interval from 2 to 3

== Examples of Use of Distance from Subset of Real Numbers to Element ==
Consider the open real interval $I = \openint 2 3$.
The real numbers $2$ and $3$ are both zero distance from $I$ but are not elements of $I$.
== Sources ==
* : $\S 2$: Continuum Property: Exercise $\S 2.13 \ (5) \ \text {(i)}$
Category:Examples of Distance from Subset of Real Numbers to Element
Distance from Subset of Real Numbers to Element/Proof 1

== Theorem ==
Let $S$ be a subset of the set of real numbers $\R$.
Let $x \in \R$ be a real number.
Let $\map d {x, S}$ be the distance between $x$ and $S$.
Then:
== Proof ==
From the definition of distance:
:$\forall x, y \in \R: \map d {x, y} = \size {x - y}$
Thus:
:$\ds \map d {x, S} = \map {\inf_{y \mathop \in S} } {\size {x - y} }$
Consider the set $T = \set {\size {x - y}: y \in S}$.
This has $0$ as a lower bound as Absolute Value is Bounded Below by Zero.
So:
:$\ds \map d {x, S} = \map {\inf_{y \mathop \in S} } {\size {x - y} } \ge 0$
If $x \in S$ then:
:$\size {x - x} = 0 \in T$
and so:
:$\ds 0 \le \map {\inf_{y \mathop \in S} } {\map d {x, y} }$
Thus:
:$\ds \map d {x, S} = \map {\inf_{y \mathop \in S} } {\map d {x, y} } = 0$
== Sources ==
* : $\S 2$: Continuum Property: Exercise $\S 2.13 \ (5) \ \text {(i)}$
Category:Distance from Subset of Real Numbers to Element
Distance from Subset of Real Numbers to Element/Proof 2

== Theorem ==
Let $S$ be a subset of the set of real numbers $\R$.
Let $x \in \R$ be a real number.
Let $\map d {x, S}$ be the distance between $x$ and $S$.
Then:
== Proof ==
Recall from Real Number Line is Metric Space that the set of real numbers $\R$ with the distance function $d$ is a metric space.
The result is then seen to be an example of Distance from Subset to Element.
Category:Distance from Subset of Real Numbers to Element
Distance from Subset of Real Numbers to Infimum

== Theorem ==
Let $S$ be a subset of the set of real numbers $\R$.
Let $x \in \R$ be a real number.
Let $\map d {x, S}$ be the distance between $x$ and $S$.
Let $S$ be bounded below such that $\xi = \inf S$.
Then:
:$\map d {\xi, S} = 0$
== Proof 1 ==
== Proof 2 ==
Category:Distance Function
Category:Distance from Subset of Real Numbers to Infimum
Distance from Subset of Real Numbers to Infimum/Proof 1

== Theorem ==
Let $S$ be a subset of the set of real numbers $\R$.
Let $x \in \R$ be a real number.
Let $\map d {x, S}$ be the distance between $x$ and $S$.
== Proof ==
From the definition of distance:
:$\forall x, y \in \R: \map d {x, y} = \size {x - y}$
Thus:
:$\ds \map d {x, S} = \map {\inf_{y \mathop \in S} } {\size {x - y} }$
Let $\xi = \inf S$.
Consider $\map d {-\xi, S'}$ where $S' = \set {-\xi: \xi \in S}$.
By Negative of Infimum is Supremum of Negatives:
:$\xi = \inf S \implies -\xi = \sup S'$
Thus from the above, $\map d {-\xi, S'} = 0$ and hence the result.
== Sources ==
* : $\S 2$: Continuum Property: Exercise $\S 2.13 \ (5) \ \text {(ii)}$
Category:Distance from Subset of Real Numbers to Infimum
Distance from Subset of Real Numbers to Infimum/Proof 2

== Theorem ==
Let $S$ be a subset of the set of real numbers $\R$.
Let $x \in \R$ be a real number.
Let $\map d {x, S}$ be the distance between $x$ and $S$.
== Proof ==
Recall from Real Number Line is Metric Space that the set of real numbers $\R$ with the distance function $d$ is a metric space.
The result is then seen to be an example of Distance from Subset to Infimum.
Category:Distance from Subset of Real Numbers to Infimum
Distance from Subset of Real Numbers to Supremum

== Theorem ==
Let $S$ be a subset of the set of real numbers $\R$.
Let $x \in \R$ be a real number.
Let $\map d {x, S}$ be the distance between $x$ and $S$.
Let $S$ be bounded above such that $\xi = \sup S$.
Then:
:$\map d {\xi, S} = 0$
== Proof 1 ==
== Proof 2 ==
Category:Distance Function
Category:Distance from Subset of Real Numbers to Supremum
Distance from Subset of Real Numbers to Supremum/Proof 1

== Theorem ==
Let $S$ be a subset of the set of real numbers $\R$.
Let $x \in \R$ be a real number.
Let $\map d {x, S}$ be the distance between $x$ and $S$.
== Proof ==
From the definition of distance:
:$\forall x, y \in \R: \map d {x, y} = \size {x - y}$
Thus:
:$\ds \map d {x, S} = \map {\inf_{y \mathop \in S} } {\size {x - y} }$
Let $\xi = \sup S$.
Then:
:$\forall y \in S: \size {\xi - y} = \xi - y$
So we need to show that no $h > 0$ can be a lower bound for $T = \set {\size {\xi - y}: y \in S}$.
 $\exists h > 0: \forall y \in S: \xi - y \ge h$.
But then:
:$\forall y \in S: y \le \xi - h$
and hence $\xi - h$ is an upper bound for $T$ smaller than $\xi = \sup S$.
But this contradicts the definition of supremum, that is the smallest upper bound.
So there is no such $h > 0$.
Hence by Proof by Contradiction it follows that:
:$\map d {\xi, S} = 0$.
== Sources ==
* : $\S 2$: Continuum Property: Exercise $\S 2.13 \ (5) \ \text {(ii)}$
Category:Distance from Subset of Real Numbers to Supremum
Distance from Subset of Real Numbers to Supremum/Proof 2

== Theorem ==
Let $S$ be a subset of the set of real numbers $\R$.
Let $x \in \R$ be a real number.
Let $\map d {x, S}$ be the distance between $x$ and $S$.
== Proof ==
Recall from Real Number Line is Metric Space that the set of real numbers $\R$ with the distance function $d$ is a metric space.
The result is then seen to be an example of Distance from Subset to Supremum.
Category:Distance from Subset of Real Numbers to Supremum
Distance from Subset to Element

== Theorem ==
Let $\struct {M, d}$ be a metric space.
Let $S \subseteq M$ be a subset of $M$.
Let $s \in S$.
Then:
:$\map d {s, S} = 0$
where $\map d {s, S}$ denotes the distance between $s$ and $S$.
== Proof ==
By Distance between Element and Subset is Nonnegative:
:$\map d {s, S} \ge 0$
Also, because:
:$\map d {s, S} = 0$
and $s \in S$, it follows that:
:$\map d {s, S} \le 0$
Hence the result.
Category:Distance Function
Distance from Subset to Infimum

== Theorem ==
Let $S \subseteq \R$ be a subset of the real numbers.
Suppose that the infimum $\inf S$ of $S$ exists.
Then:
:$\map d {\inf S, S} = 0$
where $\map d {\inf S, S}$ is the distance between $\inf S$ and $S$.
== Proof ==
By Distance between Element and Subset is Nonnegative:
:$\map d {\inf S, S} \ge 0$
By definition of infimum:
:$\forall \epsilon > 0: \exists s \in S: \map d {\inf S, s} < \epsilon$
meaning that, by nature of the infimum and the definition of $\map d {\inf S, S}$:
:$\forall \epsilon > 0: \map d {\inf S, S} < \epsilon$
Together, these two observations lead to the conclusion that:
:$\map d {\inf S, S} = 0$
as desired.
Category:Real Analysis
Distance from Subset to Supremum

== Theorem ==
Let $S \subseteq \R$ be a subset of the real numbers.
Suppose that the supremum $\sup S$ of $S$ exists.
Then:
:$\map d {\sup S, S} = 0$
where $\map d {\sup S, S}$ is the distance between $\sup S$ and $S$.
== Proof ==
By Distance between Element and Subset is Nonnegative:
:$\map d {\sup S, S} \ge 0$
By definition of supremum:
:$\forall \epsilon > 0: \exists s \in S: \map d {\sup S, s} < \epsilon$
meaning that, by nature of the infimum and the definition of $\map d {\sup S, S}$:
:$\forall \epsilon > 0: \map d {\sup S, S} < \epsilon$
Together, these two observations lead to the conclusion that:
:$\map d {\sup S, S} = 0$
as desired.
== Also see ==
* Distance from Subset to Infimum
Category:Real Analysis
Category:Distance Function
Distance from Vertex of Triangle to Orthocenter

== Theorem ==
Let $\triangle ABC$ be a triangle with sides $a$, $b$ and $c$ opposite vertices $A$, $B$ and $C$ respectively.
Let $H$ be the orthocenter of $\triangle ABC$.
Then:
:$AH = 2 R \size {\cos A}$
where $R$ denotes the circumradius of $\triangle ABC$.
== Proof ==
== Sources ==
* : $\text V$. Trigonometry: Formulae $(46)$
Category:Distance from Vertex of Triangle to Orthocenter
Category:Orthocenters of Triangles
Distance from Vertex of Triangle to Orthocenter/Proof

== Theorem ==
== Proof ==
This proof is for the case when $A \le 90 \degrees$.
The proof of the obtuse case is very similar.
:
We construct the circumcircle of $\triangle ABC$, whose circumcenter is $K$ and whose circumradius is $R$.
We construct the orthocenter $H$ of $\triangle ABC$ as the intersection of the altitudes $AD$ and $BE$.
First we note the following:
Consider $\Box EHCD$.
$\angle CDH$ and $\angle CEH$ are both right angles.
Hence from Opposite Angles of Cyclic Quadrilateral sum to Two Right Angles, $\Box EHCD$ is a cyclic quadrilateral.
Thus:
Hence:
The result follows.
== Sources == 
* : $\text V$. Trigonometry: The orthocentre
Category:Distance from Vertex of Triangle to Orthocenter
Distance of Directrix of Ellipse from Center

== Theorem ==
Let $K$ be a ellipse with semi-major axis $a$ and eccentricity $e$.
The distance of the directrix of $K$ from the center of $K$ is equal to $\dfrac a e$.
== Proof ==
== Sources ==
* 
* 
Category:Directrices of Ellipses
Category:Ellipses
Distance of Point from Origin in Cartesian Coordinates

== Theorem ==
Let $P = \tuple {x, y}$ be a point in the cartesian plane.
Then $P$ is at a distance of $\sqrt {x^2 + y^2}$ from the origin.
== Proof ==
:
By definition of the cartesian plane, the point $P$ is $x$ units from the $y$-axis and $y$ units from the $x$-axis.
The $y$-axis and $x$-axis are perpendicular to each other, also by definition.
Thus $x$, $y$ and $OP$ form a right-angled triangle.
By Pythagoras' Theorem:
:$OP^2 = x^2 + y^2$
Hence the result.
== Sources ==
* : $\S 5$: Trigonometric Functions
Category:Cartesian Coordinate Systems
Distance on Real Numbers is Metric

== Theorem ==
Let $x, y \in \R$ be real numbers.
Let $\map d {x, y}$ be the distance between $x$ and $y$:
:$\map d {x, y} = \size {x - y}$
Then $\map d {x, y}$ is a metric on $\R$.
Thus it follows that $\tuple {\R, d}$ is a metric space.
== Proof ==
We check the metric space axioms in turn.
=== ===
The statement of this axiom is:
:: $\forall x \in X: \size {x - x} = 0$
This follows from the definition of absolute value.
=== ===
The statement of this axiom is:
:: $\forall x, y, z \in X: \size {x - y} + \size {y - z} \ge \size {x - z}$
We have:
:$\paren {x - y} + \paren {y - z} = \paren {x - z}$
The result follows from the Triangle Inequality for Real Numbers.
=== ===
The statement of this axiom is:
:: $\forall x, y \in X: \size {x - y} = \size {y - x}$
As $x - y = -\paren {y - x}$, it follows from the definition of absolute value that $\size {x - y} = \size {y - x}$.
=== ===
The statement of this axiom is:
:: $\forall x, y \in X: x \ne y \implies \size {x - y} > 0$
This follows from the definition of absolute value.
Having verified all the axioms, we conclude $d$ is a metric.
== Sources ==
* : $\S 1$: Real Numbers: Exercise $\S 1.20 \ (3)$
Category:Real Analysis
Category:Real Number Line with Euclidean Metric
Distinct Elements are Parallel iff Pair forms Circuit

== Theorem ==
Let $M = \struct {S, \mathscr I}$ be a matroid.
Let $x, y \in S : x \ne y$.
Then:
:$x$ and $y$ are parallel $\set {x, y}$ is a circuit
== Proof ==
=== Necessary Condition ===
Let $x$ and $y$ be parallel.
By definition of parallel:
:$\set x$ is independent 
:$\set y$ is independent 
:$\set {x, y}$ is dependent
Let $A \subseteq \set {x, y}$ be dependent.
Thus:
:$A \ne \set x, \set y$
By matroid axiom $(\text I 1)$:
:$\O$ is independent
Thus:
:$A \ne \O$
From Power Set of Doubleton:
:$\powerset {\set {x, y} } = \set {\O, \set x, \set y, \set {x, y} }$
Thus:
:$A = \set {x, y}$
By definition of a minimal set:
:$\set {x, y}$ is a minimal dependent subset
It follows that $\set {x, y}$ is a circuit by definition.
=== Sufficient Condition ===
Let $\set {x, y}$ by a circuit.
By definition of a circuit:
:$\set {x, y}$ is a minimal dependent subset
By definition of a subset:
:$\set x, \set y \subseteq \set {x, y}$
By definition of a minimal dependent subset: 
:$\set x, \set y$ are independent subsets
It follows that $x$ and $y$ are parallel by definition.
== Sources ==
* Chapter $1.$ $\S 4.$ Loops and parallel elements
Category:Matroid Parallelism
Category:Matroid Circuits
Distinct Lower Sections of Well-Ordered Class are not Order Isomorphic

== Theorem ==
Let $\struct {A, \preccurlyeq}$ be a well-ordered class.
Let $L_1$ and $L_2$ be distinct lower sections of $\struct {A, \preccurlyeq}$.
Then $L_1$ and $L_2$ are not order isomorphic $\preccurlyeq$.
== Proof ==
A lower section of $A$ is a subclass of $A$.
Hence by definition of well-ordered class. $L_1$ and $L_2$ are themselves well-ordered classes.
We have $L_1 \ne L_2$.
The result follows from 
== Sources ==
* : Chapter $6$: Order Isomorphism and Transfinite Recursion: $\S 2$ Isomorphisms of well orderings: Corollary $2.6$
Category:Well-Orderings
Category:Lower Sections
Category:Order Isomorphisms
Distinct Matroid Elements are Parallel iff Each is in Closure of Other

== Theorem ==
Let $M = \struct {S, \mathscr I}$ be a matroid.
Let $\sigma: \powerset S \to \powerset S$ denote the closure operator of $M$.
Let $x, y \in S : x \ne y$.
Then $x$ is parallel to $y$ :
:$(1)\quad x$ and $y$ are not loops
:$(2)\quad x \in \map \sigma {\set y}$
:$(3)\quad y \in \map \sigma {\set x}$
== Proof ==
=== Lemma ===
=== Necessary Condition ===
Let $x$ and $y$ be parallel.
By definition of parallel:
:$\set x$ and $\set y$ are independent 
:$\set {x, y}$ is dependent
By definition of a loop:
:$x$ and $y$ are not loops 
From Lemma:
:$x \in \map \sigma {\set y}$
:$y \in \map \sigma {\set x}$
It has been shown that conditions $(1), (2)$ and $(3)$ above hold.
=== Sufficient Condition ===
Let conditions $(1), (2)$ and $(3)$ above hold.
By definition of a loop:
:$\set x$ and $\set y$ are independent 
From Lemma:
:$\set {y, x} \notin \mathscr I$
It follows that $x$ is parallel to $y$ by definition. 
== Sources ==
* Chapter $1.$ $\S 4.$ Loops and parallel elements
Category:Matroid Closure
Category:Matroid Parallelism
Category:Distinct Matroid Elements are Parallel iff Each is in Closure of Other
Distinct Matroid Elements are Parallel iff Each is in Closure of Other/Lemma

== Theorem ==
Let $M = \struct {S, \mathscr I}$ be a matroid.
Let $a, b \in S$.
Let $\set a$ and $\set b$ be independent.
Then $\set {a, b}$ is dependent :
:$a \in \map \sigma {\set b}$
and
:$b \in \map \sigma {\set a}$
where $\sigma: \powerset S \to \powerset S$ denotes the closure operator of $M$.
== Proof ==
Category:Distinct Matroid Elements are Parallel iff Each is in Closure of Other
Distinct Ordinals are not Order Isomorphic

== Theorem ==
Let $\alpha$ and $\beta$ be ordinals such that $\alpha \ne \beta$.
Then $\alpha$ and $\beta$ are not order isomorphic.
== Proof ==
By definition, an ordinal is well-ordered by the subset relation.
From Class of All Ordinals is Well-Ordered by Subset Relation, the class of all ordinals is a nest.
Hence:
:$\paren {\alpha \subsetneqq \beta} \lor \paren {\beta \subsetneqq \alpha}$
The result follows by Well-Ordered Class is not Isomorphic to Initial Segment.
== Sources ==
* : Chapter $6$: Order Isomorphism and Transfinite Recursion: $\S 2$ Isomorphisms of well orderings: Corollary $2.5$
Category:Ordinals
Category:Well-Orderings
Category:Initial Segments
Distinct Points in Metric Space have Disjoint Neighborhoods

== Theorem ==
Let $M = \struct {A, d}$ be a metric space.
Let $x, y \in M: x \ne y$.
Then there exist neighborhoods $N_x$ and $N_y$ of $x$ and $y$ respectively such that $N_x \cap N_y = \O$, that is, that are disjoint.
== Proof ==
Let $x, y \in A: x \ne y$.
From Distinct Points in Metric Space have Disjoint Open Balls, there exist disjoint open $\epsilon$-balls $\map {B_\epsilon} x$ and $\map {B_\epsilon} y$ containing $x$ and $y$ respectively.
From Open Ball is Neighborhood of all Points Inside it follows that $\map {B_\epsilon} x$ and $\map {B_\epsilon} y$ are neighborhoods of $x$ and $y$ respectively.
Hence the result.
== Sources ==
* : Chapter $2$: Metric Spaces: $\S 4$: Open Balls and Neighborhoods: Exercise $6$
Category:Neighborhoods
Category:Metric Spaces
Distinct Points in Metric Space have Disjoint Open Balls

== Theorem ==
Let $M = \struct {A, d}$ be a metric space.
Let $x, y \in M: x \ne y$.
Then there exist disjoint open $\epsilon$-balls $\map {B_\epsilon} x$ and $\map {B_\epsilon} y$ containing $x$ and $y$ respectively.
== Proof 1 ==
== Proof 2 ==
Category:Open Balls
Category:Distinct Points in Metric Space have Disjoint Open Balls
Distinct Points in Metric Space have Disjoint Open Balls/Proof 1

== Theorem ==
== Proof ==
Let $\map d {x, y} = 2 \epsilon$.
Let $\map {B_\epsilon} x$ and $\map {B_\epsilon} y$ denote the open $\epsilon$-balls of $x$ and $y$ in $M$.
From Open Balls whose Distance between Centers is Twice Radius are Disjoint, $\map {B_\epsilon} x$ and $\map {B_\epsilon} y$ are the disjoint open $\epsilon$-balls whose existence we are to demonstrate.
Category:Distinct Points in Metric Space have Disjoint Open Balls
Distinct Points in Metric Space have Disjoint Open Balls/Proof 2

== Theorem ==
== Proof ==
Let $x, y \in A: x \ne y$.
Then $\map d {x, y} > 0$.
Put $\epsilon = \dfrac {\map d {x, y} } 2$.
Let $\map {B_\epsilon} x$ and $\map {B_\epsilon} y$ be the open $\epsilon$-balls of $x$ and $y$ respectively.
Suppose $\map {B_\epsilon} x$ and $\map {B_\epsilon} y$ are not disjoint.
Then $\exists z \in M$ such that $z \in \map {B_\epsilon} x$ and $z \in \map {B_\epsilon} y$.
Then $\map d {x, z} < \epsilon$ and $\map d {z, y} < \epsilon$.
Hence $\map d {x, z} + \map d {z, y} < 2 \epsilon = \map d {x, y}$.
This contradicts the definition of a metric, so there can be no such $z$.
Hence the open balls must be disjoint.
Category:Distinct Points in Metric Space have Disjoint Open Balls
Distributed Term of Categorical Syllogism/Examples

== Definition ==
=== Universal Affirmative ===
=== Universal Negative ===
=== Particular Affirmative ===
=== Particular Negative ===
This can be tabulated as follows:
:$\begin{array}{rcl}
\mathbf A & (S, & P) \\
& d & u
\end{array}
\qquad
\begin{array}{rcl}
\mathbf E & (S, & P) \\
& d & d
\end{array}$
:$\begin{array}{rcl}
\mathbf I & (S, & P) \\
& u & u
\end{array}
\qquad
\begin{array}{rcl}
\mathbf O & (S, & P) \\
& u & d
\end{array}$
where $d$ denotes a distributed term and $u$ denotes an undistributed term.
== Sources ==
* : Chapter $4$: The Predicate Calculus $2$: $4$ The Syllogism
Category:Examples of Distributed Terms of Categorical Syllogisms
Distributed Term of Categorical Syllogism/Examples/Particular Affirmative

== Definition ==
Consider the particular affirmative categorical statement $\map {\mathbf I} {S, P}$.
Both the subject $S$ and the predicate $P$ of $\map {\mathbf I} {S, P}$ are undistributed.
== Sources ==
* : Chapter $4$: The Predicate Calculus $2$: $4$ The Syllogism
Category:Examples of Distributed Terms of Categorical Syllogisms
Distributed Term of Categorical Syllogism/Examples/Particular Negative

== Definition ==
Consider the particular negative categorical statement $\map {\mathbf O} {S, P}$.
The subject $S$ of $\map {\mathbf O} {S, P}$ is undistributed.
The predicate $P$ of $\map {\mathbf O} {S, P}$ is distributed.
== Sources ==
* : Chapter $4$: The Predicate Calculus $2$: $4$ The Syllogism
Category:Examples of Distributed Terms of Categorical Syllogisms
Distributed Term of Categorical Syllogism/Examples/Universal Affirmative

== Definition ==
Consider the universal affirmative categorical statement $\map {\mathbf A} {S, P}$.
The subject $S$ of $\map {\mathbf A} {S, P}$ is distributed.
The predicate $P$ of $\map {\mathbf A} {S, P}$ is undistributed.
== Sources ==
* : Chapter $4$: The Predicate Calculus $2$: $4$ The Syllogism
Category:Examples of Distributed Terms of Categorical Syllogisms
Distributed Term of Categorical Syllogism/Examples/Universal Negative

== Definition ==
Consider the universal negative categorical statement $\map {\mathbf E} {S, P}$.
Both the subject $S$ and the predicate $P$ of $\map {\mathbf E} {S, P}$ are distributed.
== Sources ==
* : Chapter $4$: The Predicate Calculus $2$: $4$ The Syllogism
Category:Examples of Distributed Terms of Categorical Syllogisms
Distributed Term of Conclusion of Valid Categorical Syllogism is Distributed in Premise

== Theorem ==
Let $T$ be a term of the conclusion $C$ of a valid categorical syllogism $Q$.
Let $T$ be distributed in $C$.
Then $T$ is also distributed in whichever premise of $Q$ in which it appears.
== Proof ==
== Also see ==
* Rules of Quantity
* Middle Term of Valid Categorical Syllogism is Distributed at least Once
== Historical Note ==
== Sources ==
* : Chapter $4$: The Predicate Calculus $2$: $4$ The Syllogism: Exercises: $\text{A}$: Rules of Quantity: $1$
Category:Rules of Quantity
Category:Distributed Terms of Categorical Syllogisms
Distribution Function of Finite Borel Measure is Bounded

== Theorem ==
Let $\mu$ be a finite Borel measure on $\R$.
Let $F_\mu$ be the distribution function of $\mu$.
Then $F_\mu$ is bounded.
== Proof ==
Let $x \in \R$.
Then: 
:$\O \subseteq \hointl {-\infty} x \subseteq \R$
Then, from Measure is Monotone we have: 
:$\map \mu \O \le \map \mu {\hointl {-\infty} x} \le \map \mu \R$
From Empty Set is Null Set, we have: 
:$\map \mu \O = 0$
Since $\mu$ is a finite measure, we have: 
:$\map \mu \R < \infty$
So, we have: 
:$0 \le \map \mu {\hointl {-\infty} x} \le \map \mu \R < \infty$
So:
:$0 \le \map {F_\mu} x \le \map \mu \R$ for each $x \in \R$. 
== Sources ==
* : $1.3$: Outer Measures
Category:Distribution Function of Finite Borel Measure
Distribution Function of Finite Borel Measure is Continuous at Point iff Measure Continuous at Point

== Theorem ==
Let $\mu$ be a finite Borel measure.
Let $F_\mu$ be the distribution function of $\mu$.
Let $x \in \R$.
Then $F_\mu$ is continuous at $x$ $\map \mu {\set x} = 0$.
== Proof ==
From Distribution Function of Finite Borel Measure is Right-Continuous, $F_\mu$ is right-continuous at $x$.
Hence from Continuous at Point iff Left-Continuous and Right-Continuous, the claim is equivalent to:
:$F_\mu$ is left-continuous at $x$ $\map \mu {\set x} = 0$.
That is:
:$\ds \lim_{y \to x^-} \map {F_\mu} y = \map {F_\mu} x$
:
:$\map \mu {\set x} = 0$ 
From Left Limit of Distribution Function of Finite Borel Measure, this is equivalent to:
:$\map \mu {\hointl {-\infty} x} = \map \mu {\openint {-\infty} x}$ $\map \mu {\set x} = 0$.
From Measure of Set Difference with Subset, we indeed have that:
:$\map \mu {\hointl {-\infty} x} = \map \mu {\openint {-\infty} x}$ $\map \mu {\hointl {-\infty} x} - \map \mu {\openint {-\infty} x} = \map \mu {\set x} = 0$
as required.
Category:Distribution Function of Finite Borel Measure
Distribution Function of Finite Borel Measure is Increasing

== Theorem ==
Let $\mu$ be a finite Borel measure on $\R$.
Let $F_\mu$ be the distribution function of $\mu$.
Then $F_\mu$ is an increasing function.
== Proof ==
Let $x, y \in \R$ be such that $x \le y$.
Then: 
:$\hointl {-\infty} x \subseteq \hointl {-\infty} x$
So, from Measure is Monotone:
:$\map \mu {\hointl {-\infty} x} \le \map \mu {\hointl {-\infty} y}$
That is: 
:$\map {F_\mu} x \le \map {F_\mu} y$ whenever $x \le y$.
So $F_\mu$ is an increasing function.
== Sources ==
* : $1.3$: Outer Measures
Category:Distribution Function of Finite Borel Measure