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Migrate to KaTeX

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Nathan Nguyen
2025-02-07 21:38:28 -06:00
parent 41af035e50
commit c7c46c56ee
27 changed files with 369 additions and 391 deletions
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4
docs/math/abstract-algebra/16-rings.md
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@@ -155,13 +155,13 @@ $$
\end{align}
$$
**Theorem**. Chineese Remainer Theorem. Let $n_1, \ldots, n_k \in \mathbb{N}$ be given such that $\gcd(n_i, n_j) = 1$. Then, for any integers $a_1, \ldots, a_k$, the system
**Theorem**. Chinese Remainder Theorem. Let $n_1, \ldots, n_k \in \mathbb{N}$ be given such that $\gcd(n_i, n_j) = 1$. Then, for any integers $a_1, \ldots, a_k$, the system
$$
\begin{align}
x &\equiv a_1 \pmod{n_1} \\
x &\equiv a_2 \pmod{n_2} \\
\vdots
\ldots
x &\equiv a_k \pmod{n_k}
\end{align}
$$

2
docs/math/abstract-algebra/18-integral-domains.md
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@@ -53,7 +53,7 @@ Additionally, $F_D$ is unique. That is, given field $E$ such that $E \supset D$,
**Lemma**. 18.14: Let $D$ be a PID. Let $I_1 \subseteq I_2 \subseteq \ldots$. Then, there exists some integer $N$ such that $I_n = I_N$ for all $n > N$. That is, any chain of ideals converges.
**Definition**. Any commutative ring that satisfies the above condition (the *ascending chain condition*), even if it's not a PID, is called a *Noetherien ring*.
**Definition**. Any commutative ring that satisfies the above condition (the *ascending chain condition*), even if it's not a PID, is called a *Noetherian ring*.
**Theorem**. 18.15: Every PID is a UFD. Note that the converse is not true.

18
docs/math/abstract-algebra/DF-10-modules.md
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@@ -5,7 +5,7 @@
**Definition**. Let $R$ be a ring. A *left $R$-module* or a *left module over $R$* is a nonempty set $M$ together with
1. A binary operation $+$ on $M$ under which $M$ is an abelian group
2. An action $\cross$ of $R$ on $M$, that is, a map or function $R \cross M \rightarrow M$, denoted $rm$, that for all $r, s \in R, m, n \in M$ satisfies
2. An action $\times$ of $R$ on $M$, that is, a map or function $R \times M \rightarrow M$, denoted $rm$, that for all $r, s \in R, m, n \in M$ satisfies
- $(r + s)m = rm + sm$
- $(rs)m = r(sm)$
- $r(m + n) = rm + rn$
@@ -21,12 +21,12 @@
---
**Example**. Let $F$ be a field and $F[x]$ a polynomial ring. Then, let $V$ be a vector space of $F$, and $T$ be a linear transformation from $V$ to itself. That is, $V: T \rightarrow T$. We know that $V$ is an $F$-module. We will want to show that $V$ can be written as an $F[x]$-module for some choice of $T$. That is, we want an action $F[x] \cross V \rightarrow V$.
**Example**. Let $F$ be a field and $F[x]$ a polynomial ring. Then, let $V$ be a vector space of $F$, and $T$ be a linear transformation from $V$ to itself. That is, $V: T \rightarrow T$. We know that $V$ is an $F$-module. We will want to show that $V$ can be written as an $F[x]$-module for some choice of $T$. That is, we want an action $F[x] \times V \rightarrow V$.
Now, for a given linear transformation $T$, consider some polynomial $p(x) = a_n x^n + \ldots + a_0$ and some $v \in V$. We define $p(x) \cross v$ by$
Now, for a given linear transformation $T$, consider some polynomial $p(x) = a_n x^n + \ldots + a_0$ and some $v \in V$. We define $p(x) \times v$ by$
$$
p(x) \cross v = a_n T^n(v) + a_{n-1} T^{n-1}(v) + \ldots + a_0 v
p(x) \times v = a_n T^n(v) + a_{n-1} T^{n-1}(v) + \ldots + a_0 v
$$
with $T^n$ being defined as applying $T$ a total of $n$ times.
@@ -50,20 +50,20 @@ with $T^n$ being defined as applying $T$ a total of $n$ times.
**Theorem**. An $R$-module homomorphism is an *isomorphism* if it is 1-1 and onto, and said modules are *isomorphic*.
**Definition**. Let $M, N$ be $R$-modules. The set $\Hom_R(M, N)$ is the set of all homomorphisms from $M$ to $N$.
**Definition**. Let $M, N$ be $R$-modules. The set $\text{Hom}_R(M, N)$ is the set of all homomorphisms from $M$ to $N$.
**Proposition**. Let $M$, $N$, and $L$ be $R$-modules. Then,
1. A function $\varphi: M \rightarrow N$ is an $R$-module homomorphism if and only if $\varphi(rx + y) = r\varphi(x) + \varphi(y)$ for all $x, y \in M$ and $r \in R$.
2. Let $\varphi, \psi \in \Hom_R(M, N)$. Then, define $\varphi + \psi$ as
2. Let $\varphi, \psi \in \text{Hom}_R(M, N)$. Then, define $\varphi + \psi$ as
$$
(\varphi + \psi)(m) = \varphi(m) + \psi(m)
$$
Then, $\varphi + \psi \in \Hom_R(M, N)$. Additionally, if $R$ is commutative, with $(r\varphi)(m) = r(\varphi(m))$, then $r\varphi \in \Hom_R(M,N)$
3. If $\varphi \in \Hom_R(L, M)$ and $\psi \in \Hom_R(M, N)$, then $\psi \circ \varphi \in \Hom_R(L, N)$
4. $\Hom_R(M, M)$ is a ring with identity. With $R$ being commutative, $\Hom_R(M, M)$ is an $R$-algebra.
Then, $\varphi + \psi \in \text{Hom}_R(M, N)$. Additionally, if $R$ is commutative, with $(r\varphi)(m) = r(\varphi(m))$, then $r\varphi \in \text{Hom}_R(M,N)$
3. If $\varphi \in \text{Hom}_R(L, M)$ and $\psi \in \text{Hom}_R(M, N)$, then $\psi \circ \varphi \in \text{Hom}_R(L, N)$
4. $\text{Hom}_R(M, M)$ is a ring with identity. With $R$ being commutative, $\text{Hom}_R(M, M)$ is an $R$-algebra.
**Proposition**. Let $R$ be a ring, $M$ an $R$-module, and $N \subseteq M$ an $R$-submodule. then, $M/N$ can be made into an $R$-module by defining addition. With $r \in R$ and $x + N \in M/N$,

16
docs/math/abstract-algebra/DF-12-modules-pids.md
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@@ -25,22 +25,22 @@ there exists some $k \in \mathbb{N}$ such that given any $n \in \mathbb{N}$ with
**Definition**. Given $R$ an integral domain and $M$ an $R$-module,
$$
\Tor(M) = \{ x \in M | rx = 0 \text{ for any } r \neq 0 \}
\text{Tor}(M) = \{ x \in M | rx = 0 \text{ for any } r \neq 0 \}
$$
This is the *torsion submodule* of $M$. If $\Tor(M)$ is empty, then $M$ is *torsion-free*.
This is the *torsion submodule* of $M$. If $\text{Tor}(M)$ is empty, then $M$ is *torsion-free*.
**Definition** Let $R$ be an integral domain and $M$ be an $R$-module. Then, given a submodule $N$,
$$
\Ann_R(N) = \{r \in R | rn = 0 \text{ for all } n \in N \}
\text{Ann}_R(N) = \{r \in R | rn = 0 \text{ for all } n \in N \}
$$
This ideal of $R$ is the *annihilator of $N$*. That is, $\Ann(N)$ is the set of elements of $R$ such that $(r)N = \{ 0 \}$.
This ideal of $R$ is the *annihilator of $N$*. That is, $\text{Ann}(N)$ is the set of elements of $R$ such that $(r)N = \{ 0 \}$.
Note that if $N$ is not a torsion submodule of $M$, then $\Ann(N) = (0)R$. Additionally, given $N, L$ are submodules of $M$ with $N \subseteq L$, then $\Ann(N) \subseteq \Ann(L)$.
Note that if $N$ is not a torsion submodule of $M$, then $\text{Ann}(N) = (0)R$. Additionally, given $N, L$ are submodules of $M$ with $N \subseteq L$, then $\text{Ann}(N) \subseteq \text{Ann}(L)$.
Additionally, if $R$ is a PID, as $\Ann_R(N)$ is an ideal, $\Ann(N) = (n)R$ and $\Ann(L) = (l)R$ for some $n, l \in R$ such that $n | l$.
Additionally, if $R$ is a PID, as $\text{Ann}_R(N)$ is an ideal, $\text{Ann}(N) = (n)R$ and $\text{Ann}(L) = (l)R$ for some $n, l \in R$ such that $n | l$.
**Definition**. Given any integral domain $R$, the *rank* of an $R$-module $M$ is the maximum number of $R$-linearly independent elements of M.
@@ -64,7 +64,7 @@ $$
- Note that
$$
\Tor{M} \cong \frac{R}{(a_1)R} \oplus \frac{R}{(a_2)R} \oplus \ldots \oplus \frac{R}{(a_m)R}
\text{Tor}{M} \cong \frac{R}{(a_1)R} \oplus \frac{R}{(a_2)R} \oplus \ldots \oplus \frac{R}{(a_m)R}
$$
As a consequence, $M$ is a torsion module if and only if $r = 0$.
@@ -79,4 +79,4 @@ $$
with $p_t$ non-unique primes and $\alpha_t$ non-unique, but with $(p_t^{\alpha_t})$ unique. These are called the *elementary divisors* of $M$.
TODO: Incomplete for Now
TODO: Incomplete for Now

4
docs/math/abstract-algebra/DF-13-fields.md
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@@ -1,8 +1,8 @@
# Dummit & Foote Chapter 12 - Field Theory
# Dummit & Foote Chapter 13 - Field Theory
## Section 13.1 Basic Theory of Field Extensions
**Definition**. The *charactaristic* of a field $F$ is the smallest positive integer $p$ such that $1_F * p = 0$. It follows that $p$ is $0$ or prime, and $p \alpha = 0$ for any $\alpha \in F$.
**Definition**. The *characteristic* of a field $F$ is the smallest positive integer $p$ such that $1_F * p = 0$. It follows that $p$ is $0$ or prime, and $p \alpha = 0$ for any $\alpha \in F$.
**Definition**. If $K, F$ are fields such that $F \subseteq K$, then $K$ is an *extension field* or *extension* of $F$, denoted $K / F$.

2
docs/math/diffeq/2-1st-order.md
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@@ -103,7 +103,7 @@ $$
Since $y$ is really $y(x)$, we can make the following substitution:
$$
u = y(x) \text{ and } du = y'(x)dx = \frac{dy}{dx}{dx}
u = y(x) \text{ and } du = y'(x)dx = \frac{dy}{dx} dx
$$
This reduces the integral to the following:

20
docs/math/real-analysis/11-metric-spaces.md
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@@ -24,19 +24,19 @@ $$
---
**Definition**. Let $(S, d)$ be a metric space. Then, for each $\epsilon > 0$, the *$\epsilon$-neighborhood* or *$\epsilon$-ball* of a point $a \in S$ is the set
**Definition**. Let $(S, d)$ be a metric space. Then, for each $\varepsilon > 0$, the *$\varepsilon$-neighborhood* or *$\varepsilon$-ball* of a point $a \in S$ is the set
$$
V_\epsilon(a) = {x \in S | d(a, x) < \epsilon}
V_\varepsilon(a) = {x \in S | d(a, x) < \varepsilon}
$$
**Definition**. Let $(S, d)$ be a metric space. Then, a subset $G \subseteq S$ is *open* if for each $x \in G$, there exists some $\epsilon > 0$ so that $V_\epsilon(x) \subseteq G$.
**Definition**. Let $(S, d)$ be a metric space. Then, a subset $G \subseteq S$ is *open* if for each $x \in G$, there exists some $\varepsilon > 0$ so that $V_\varepsilon(x) \subseteq G$.
**Definition**. Let $(S, d)$ be a metric space. Then, a subset $G \subseteq S$ is *closed* if its complement $C(G) = S - G = S \ F$ is closed.
**Definition**. Let $(S, d)$ be a metric space. A point $c \in S$ is a *cluster point$ of a set $A \subseteq S$ if every $\epsilon$-neighborhood of $c$ contains some point $a \in A$ such that $a \neq c$.
**Definition**. Let $(S, d)$ be a metric space. A point $c \in S$ is a *cluster point$ of a set $A \subseteq S$ if every $\varepsilon$-neighborhood of $c$ contains some point $a \in A$ such that $a \neq c$.
**Theorem**. Every $\epsilon$-neighborhood of a point is an open set.
**Theorem**. Every $\varepsilon$-neighborhood of a point is an open set.
**Theorem**. The union of an arbitrary collection of open sets is open.
@@ -50,22 +50,22 @@ $$
---
**Definition**. A *sequence* $(x_n)$ in a metric space $(S, d)$ converges to a point $x \in S$ if given any $\epsilon > 0$, there exists a $K \in \mathbb{N}$ such that given $n \in \mathbb{N}$,
**Definition**. A *sequence* $(x_n)$ in a metric space $(S, d)$ converges to a point $x \in S$ if given any $\varepsilon > 0$, there exists a $K \in \mathbb{N}$ such that given $n \in \mathbb{N}$,
$$
n \geq K \Rightarrow d(x_n, x) \leq \epsilon
n \geq K \Rightarrow d(x_n, x) \leq \varepsilon
$$
**Theorem**. Let $(x_n)$ be a sequence in metric space $(S, d)$. Then,
- $(x_n)$ converges to $x$ if and only if every $\epsilon$-neighborhood of $x$ contains all but finitely many terms of $(x_n)$.
- $(x_n)$ converges to $x$ if and only if every $\varepsilon$-neighborhood of $x$ contains all but finitely many terms of $(x_n)$.
- If $(x_n) \rightarrow x$ and $(x_n) \rightarrow x'$, then $x = x'$.
- If $(x_n)$ converges, then $(x_n)$ is bound.
**Definition**. A sequence $(x_n)$ in metric space $(S, d)$ is a *Cauchy sequence* if for every $\epsilon > 0$, there exists some $H \in \mathbb{N}$ such that for any $m, n \in \mathbb{N}$,
**Definition**. A sequence $(x_n)$ in metric space $(S, d)$ is a *Cauchy sequence* if for every $\varepsilon > 0$, there exists some $H \in \mathbb{N}$ such that for any $m, n \in \mathbb{N}$,
$$
m, n \geq H \Rightarrow d(x_n, x_m) < \epsilon
m, n \geq H \Rightarrow d(x_n, x_m) < \varepsilon
$$
**Definition**. A metric space in which every Cauchy sequence converges is said to be *complete*.

20
docs/math/real-analysis/2-reals.md
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@@ -56,32 +56,32 @@
2. $|ab| = |a||b|$
3. $|a + b| \leq |a| + |b|$
**Corollary**. Given $a, b \in \mathbb{R}$, then $\abs{\abs{a} - \abs{b}} \leq \abs{a - b}$.
**Corollary**. Given $a, b \in \mathbb{R}$, then $||a| - |b|| \leq |a - b|$.
**Remark**. Every field has at least one absolute value function.
**Theorem**. In an ordered field $F$, for any $r > 0$, we know that
1. $\abs{x = r}$ if and only if $x = r$ or $x = -r$
2. $\abs{x < r}$ if and only if $-r < x < r$
3. $\abs{x > r}$ if either $x > r$ or $x < -r$
1. $|x = r$ if and only if $x = r$ or $x = -r$
2. $|x < r$ if and only if $-r < x < r$
3. $|x > r$ if either $x > r$ or $x < -r$
---
**Definition**. The *standard distance function* or *metric* on the real numbers $\mathbb{R}$ given $a, b$ is $\abs{a - b}$.
**Definition**. The *standard distance function* or *metric* on the real numbers $\mathbb{R}$ given $a, b$ is $|a - b|$.
**Theorem**. For any real numbers $a, b, c$,
1. $\abs{a - b} > 0$ if and only if $a \neq b$ and $\abs{a - b} = 0$ if and only if $a = b$
2. $\abs{a - b} = \abs{b - a}$
3. $\abs{a - c} \leq \abs{a - b} + \abs{b + c}$
1. $|a - b| > 0$ if and only if $a \neq b$ and $|a - b| = 0$ if and only if $a = b$
2. $|a - b| = |b - a|$
3. $|a - c| \leq |a - b| + |b + c|$
**Definition** A set together with a function satisfying these three properties is known as a *metric space*.
**Definition** The $\epsilon$-neighborhood of $a \in \mathbb{R}$, denoted $V_\epsilon(a)$ is the set of all real numbers $x \in \mathbb{R}$ such that $\abs{x - a} < \epsilon$. That is,
**Definition** The $\varepsilon$-neighborhood of $a \in \mathbb{R}$, denoted $V_\varepsilon(a)$ is the set of all real numbers $x \in \mathbb{R}$ such that $|x - a| < \varepsilon$. That is,
$$
V_\epsilon(a) = (a - \epsilon, a + \epsilon)
V_\varepsilon(a) = (a - \varepsilon, a + \varepsilon)
$$
---

22
docs/math/real-analysis/3-sequences-series.md
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@@ -26,10 +26,10 @@ $$
---
**Definition**. A sequence $X = (x_n)$ is said to *converge* to a number $x \in \mathbb{R}$ if when given any $\epsilon > 0$, there exists some $K \in \mathbb{N}$ such that for every $n \in \mathbb{N}$ with $n \geq K$,
**Definition**. A sequence $X = (x_n)$ is said to *converge* to a number $x \in \mathbb{R}$ if when given any $\varepsilon > 0$, there exists some $K \in \mathbb{N}$ such that for every $n \in \mathbb{N}$ with $n \geq K$,
$$
\abs{x_n - x} < \epsilon
|x_n - x| < \varepsilon
$$
If this is the case, we say that $X$ converges to $x$, and $x$ is a *limit* of X. This can be written as
@@ -66,9 +66,9 @@ $$
**Theorem**. Suppose $(x_n)$ is a sequence if real numbers. Then,
1. If $x_n \rightarrow x$, then $\abs{x_n} \rightarrow \abs{x}$
2. If $\abs{x_n} \rightarrow 0$, then $x_n \rightarrow 0$
3. $x_n \rightarrow x$ if and only if $\abs{x_n - n} \rightarrow 0$
1. If $x_n \rightarrow x$, then $|x_n| \rightarrow |x|$
2. If $|x_n| \rightarrow 0$, then $x_n \rightarrow 0$
3. $x_n \rightarrow x$ if and only if $|x_n - n| \rightarrow 0$
**Theorem**. Suppose $(x_n)$ is a sequence if real numbers, with each $x_n \geq 0$. Then, given some $k \in \mathbb{N}$, if $x_n \rightarrow x$, then $\sqrt[k]{x_n} \rightarrow \sqrt[k]{x}$.
@@ -100,11 +100,9 @@ is a *subsequence* of $X$,
## Section 3.5 - The Cauchy Criterion
**Definition**. A sequence $(x_n)$ is said to be a *Cauchy sequence* such that for any given $\epsilon$, there exists a natural number $H$ such that all natural numbers $m, n \geq H$, then
**Definition**. A sequence $(x_n)$ is said to be a *Cauchy sequence* such that for any given $\varepsilon$, there exists a natural number $H$ such that all natural numbers $m, n \geq H$, then
$$
\abs{x_m - x_n} \leq \epsilon
$$
$$|x_m - x_n \leq \varepsilon$$
**Theorem**. If $(x_n)$ is a Cauchy sequence, then $(x_n)$ is convergent.
@@ -130,15 +128,15 @@ $$
In particular, $s_n - s_{n - 1} = x^n$. Thus, the Cauchy criteria takes the form
**Theorem**. Cauchy Criteria for Series. The series $\sum x_n$ converges if and only if, for a given $\epsilon$, there exists some natural number $H$ such that when $m > n > H$,
**Theorem**. Cauchy Criteria for Series. The series $\sum x_n$ converges if and only if, for a given $\varepsilon$, there exists some natural number $H$ such that when $m > n > H$,
$$
\abs{s_m - s_n} = \abs{\sum_{i = m + 1}^n x_i} < \epsilon
|s_m - s_n| = |\sum_{i = m + 1}^n x_i| < \varepsilon
$$
**Corollary**. $n$-th Term Test. If $\sum x_n$ converges, then $x_n \rightarrow 0$.
**Corollary**. Absolute Convergence Test. If $\sum \abs{x_n}$ converges, then $\sum x_n$ converges.
**Corollary**. Absolute Convergence Test. If $\sum |x_n|$ converges, then $\sum x_n$ converges.
---

10
docs/math/real-analysis/4-limits.md
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@@ -18,10 +18,10 @@
---
**Definition**. Suppose $f: A \rightarrow \mathbb{R}$ is a function with domain $A \subseteq \mathbb{R}$, and let $c \in A$ be a cluster point of $A$. then, a real number $L$ is a *limit of $f$ at $c$* if given any $\epsilon > 0$, there exists some $\delta > 0$ such that
**Definition**. Suppose $f: A \rightarrow \mathbb{R}$ is a function with domain $A \subseteq \mathbb{R}$, and let $c \in A$ be a cluster point of $A$. then, a real number $L$ is a *limit of $f$ at $c$* if given any $\varepsilon > 0$, there exists some $\delta > 0$ such that
$$
0 < |x-c| < \delta \Rightarrow |f(x) - L| < \epsilon
0 < |x-c| < \delta \Rightarrow |f(x) - L| < \varepsilon
$$
**Theorem**. For a given function and cluster point, there can be at most one limit at said point.
@@ -32,12 +32,12 @@ $$
**Definition**. The *extended real numbers* are $\hat{\mathbb{R}} = \mathbb{R} \cup \{ \infty, -\infty \}$ are a totally-ordered set with supremum and infimum. Note that this set is no longer a field.
**Definition**. At any point $c$, the limit of $f$ at $c$ is infinite if given some $\alpha$, there exists some $V_\delta(c)$ such that for all $x \in V_\epsilon(c)$, then $f(x) \in V_\alpha(\infty)$.
**Definition**. At any point $c$, the limit of $f$ at $c$ is infinite if given some $\alpha$, there exists some $V_\delta(c)$ such that for all $x \in V_\varepsilon(c)$, then $f(x) \in V_\alpha(\infty)$.
**Definition**. The limit of a function at infinity is defined if for a given $\epsilon$, there exists some $\alpha$ so that there exists some $V_\delta(c)$ such that for all $x \in A$,
**Definition**. The limit of a function at infinity is defined if for a given $\varepsilon$, there exists some $\alpha$ so that there exists some $V_\delta(c)$ such that for all $x \in A$,
$$
x > \alpha \Rightarrow |f(x) - L| < \epsilon
x > \alpha \Rightarrow |f(x) - L| < \varepsilon
$$
## Section 4.2 - Limit Theorems

12
docs/math/real-analysis/5-continuity.md
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@@ -2,10 +2,10 @@
## Section 5.1 - Continuous Functions
**Definition**. Let $A \subseteq \mathbb{R}$, and $f: A \rightarrow \mathbb{R}$. Then, if $a \in A$, $f$ is *continuous at $a$* if, given any $\epsilon > 0$, there exists some $\delta > 0$ such that for all $x \in A$,
**Definition**. Let $A \subseteq \mathbb{R}$, and $f: A \rightarrow \mathbb{R}$. Then, if $a \in A$, $f$ is *continuous at $a$* if, given any $\varepsilon > 0$, there exists some $\delta > 0$ such that for all $x \in A$,
$$
|x - a| < \delta \Rightarrow |f(x) - f(a)| < \epsilon
|x - a| < \delta \Rightarrow |f(x) - f(a)| < \varepsilon
$$
Note that if $a$ is an *isolated point* of $A$, that is, not a cluster point, then $a$ is automatically continuous.
@@ -20,10 +20,10 @@ Note that a function cannot be continuous at a point outside of its domain, even
---
**Definition**. Let $(S, d_S)$ and $(T, d_T)$ be metric spaces. A function $f: S \rightarrow T$ is continuous at a point $a \in S$ if given any $\epsilon > 0$, there exists some $\delta > 0$ such that for all $x \in S$,
**Definition**. Let $(S, d_S)$ and $(T, d_T)$ be metric spaces. A function $f: S \rightarrow T$ is continuous at a point $a \in S$ if given any $\varepsilon > 0$, there exists some $\delta > 0$ such that for all $x \in S$,
$$
d_S(x, a) < \delta \Rightarrow d_T(f(x), f(a)) < \epsilon
d_S(x, a) < \delta \Rightarrow d_T(f(x), f(a)) < \varepsilon
$$
**Theorem**. A function $f: S \rightarrow T$ is continuous at a point $a \in A$ if and only if given some neighborhood $V(f(a)) \in B$, there exists some $U(a) \in A$ such that $f(U) \subseteq V$.
@@ -63,10 +63,10 @@ $$
---
**Definition**. Let $A \subseteq R$. Then, a function $f: A \rightarrow \mathbb{R}$ is *uniformly continuous* if given any $\epsilon > 0$, there exists some $\delta > 0$ depending only on $\epsilon$ such that for any $x, y \in A$,
**Definition**. Let $A \subseteq R$. Then, a function $f: A \rightarrow \mathbb{R}$ is *uniformly continuous* if given any $\varepsilon > 0$, there exists some $\delta > 0$ depending only on $\varepsilon$ such that for any $x, y \in A$,
$$
|x - y| < \delta \Rightarrow |f(x) - f(y)| < \epsilon
|x - y| < \delta \Rightarrow |f(x) - f(y)| < \varepsilon
$$
Note that if $f$ is uniformly continuous, it must be continuous on $A$.