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--- math104-s21:s:oscarxu [2021/05/11 06:41]
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 A point is in the boundary of $E$ if and only if it belongs to the closure of both $E$ and its complement.
+Series is an infinite sun $\sum_{n=1}^\infty a_n$
+Partial sum $s_n$ = $\sum_{i=1}^n a_i$
+A series converge to $\alpha$ if the corresponding partial sum converges.
+Cauchy condition for series convergence: sufficient and necessary condition
+$\forall \epsilon$ we have there exists an $N > 0$ such that $\forall n, m > N, |\sum_i={n+1}^m a_i| < |epsilon|$
+Corollary: If a series $\sum a_n$ converges, then $\lim a_n = 0$
+Comparison test:
+If $\sum a_n$ converges and $|b_n| \leq |a_n|$ for all $n$, then $\sum b_n$ converges
+If $\sum a_n = \infty$ and $|b_n| \geq |a_n|$ for all $n$, then $\sum b_n = \infty$
+Corollary: Absolutely convergent series are convergent.
+Ration test:
+A series $\sum a_n$ of nonzero terms
+converges absolutely if $\lim \sup |\frac{a_{n+1}}{a_n}| < 1$
+diverges if $\lim \inf |\frac{a_{n+1}}{a_n}| > 1$
+otherwise $\lim \inf |\frac{a_{n+1}}{a_n}| \leq 1 \leq \lim \sup |a_{n+1}/a_n|$ and the test gives no information.
+Root test:
+Let $\alpha = \lim \sup |a_n|^{\frac{1}{n}}$ Then the series $\sum a_n$
+converges absolutely if $\alpha < 1$
+diverges if $\alpha > 1$
+Alternating series test: Let $a_1 \geq a_2 \geq \cdots $ be a monotone decreasing series, $a_n \geq 0$. And assuming $\lim a_n = 0$. Then $\sum_{n=1}^\infty (-1)^{n+1}a_n = a_1 - a_2 + a_3 - a_4$ converges. The partial sums satisfy $|s - s_n| \leq a_n$ for all $n$.
+Integral test: If the terms $a_n$ in $\sum_n a_n$ are non-negative and $f(n) = a_n$ is a decreasing function on $[1, \infty]$ then let $\alpha = \lim_{n \to \infty} \int_1^n f(x) dx$
+If $\alpha = \infty$, the series diverge.
+If $\alpha < \infty$, the series converge.
+**Functions**
+Definition of function: A function from set $A$ to set $B$ is an assignment for each element $\alpha \in A$ an element $f(\alpha) \in B$.
+Injective, Surjective, Bijective
+Definition of Preimage: Given $f: A \to B$. Given a subset $E \subset B$, we have $f^{-1}(E) = \{\alpha \in A | f(\alpha) \in E\} is called the perimage of $E$ under f
+Definition of limit of a function: Suppose $p\in E'$(set of limit points of $E$), we write $f(x) \to q(\in Y)$ as $x \to p$ or $\lim_{x\to p} f(x) = q$ if $\forall \epsilon >0,\, \exists \delta >0$ such that $\forall x \isin E,\, 0<d_X(x,p)<\delta  \implies d_Y(f(x),q)<\epsilon$.
+Theorem: $\lim_{x\to p} f(x) = q$ if and only iff $\lim_{n\to\infty} f(p_n) = q$ for every sequence $(p_n)$ in $E$ such that $p_n \neq p, \lim_{n\to\infty} p_n = p$.
+Theorem: For $f,g: E \to \mathbb{R}$, suppose $p\in E;$ and $\lim_{x\to p} f(x) = A, \lim_{x\to p} g(x) = B$, then
+$\lim_{x\to p} f(x) +g(x)= A+B$
+$\lim_{x\to p} f(x)g(x) = AB$
+$\lim_{x\to p} \frac{f(x)}{g(x)} = \frac{A}{B}$ if $B\neq 0$ and $g(x)\neq 0 \, \forall x\isin E$
+$\forall c\in \mathbb{R}$, $\lim_{x\to p} c*f(x)=cA$
+Definition of pointwise continuity: Let $(X,d_X), (Y,d_Y)$ be metric spaces, $E\subset X$, $f:E \to Y$, $p \isin E$, $q=f(p)$. We say $f$ is continuous at $p$, if $\forall \epsilon>0, \exists \delta>0$ such that $\forall x\in E$ with $d_X(x,p) <\delta \implies d_Y(f(x),q)<\epsilon$.
+Theorem: Let $f:X \to \mathbb{R}^n$ with $f(x) = (f_1(x), f_2(x), ..., f_n(x))$. Then $f$ is continuous if and only if each $f_i$ is continuous.
+Definition of continuous maps:
+$f$ is continuous if and only if $\forall p\in X$, we have $\forall \epsilon >0, \exists \delta >0$ such that $f(B_{\delta}(p)) \subset B_{\epsilon}(f(p))$
+$f$ is continuous if and only if $\forall V\subset Y$ open, we have $f^{-1}(V)$ is open
+$f$ is continuous if and only if $\forall x_n \to x$ in $X$, we have $f(x_n) \rightarrow f(x)$ in $Y$
+Theorem: Given that $f$ is a continuous map from a compact metric space $X$ to another compact metric space $(Y)$, then $f(X) \subset Y$ is compact.
+Midterm 2 T/F question here:
+Let $f: X \to Y$ be a continous map between metric spaces. Let $A \subset X$ and $B \subset Y$.
+If A is open, then $f(A)$ is open. False
+If A is closed, then $f(A)$ is closed. False
+If A is bounded, then $f(A)$ is bounded. False
+If $A$ is connected, then $f(A)$ is connected. True
+If $A$ is compact, then $f(A)$ is compact. True
+If $B$ is open, then $f^{-1}(B)$ is open. True
+If $B$ is closed, then $f^{-1}(B)$ is closed. True
+If $B$ is bounded, then $f^{-1}(B)$ is bounded. False
+If $B$ is connected, then $f^{-1}(B)$ is connected. False
+If $B$ is compact, then $f^{-1}(B)$ is compact. False
+Definition of uniform continuous function: $f: X \to Y$. Suppose for all $\epsilon > 0$, we have $\sigma > 0$ such that $\forall p, q \in X$ with $d_X(p, q) < \sigma$, $d_Y(f(p), f(q)) < \epsilon$. Then $f$ is a uniform continous function.
+Theorem: Suppose $f:X \to Y$ is a continuous function between metric spaces. If $X$ is compact, then $f$ is uniformly continuous.
+Theorem: If $f:X\to Y$ is uniformly continuous and $S\subset X$ subset with induced metric, then the restriction $f|_S:S\to Y$ is uniformly continuous.
+Definition of connected: Let $X$ be a set. We say $X$ is connected if $\forall S\subset X$ we have $S$ is both open and closed, then $S$ has to be either $X$ or $\emptyset$.
+Lemma: $E$ is connected if and only if $E$ cannot be written as $A\cup B$ when $A^- \cap B = \emptyset$ and $A\cap B^- = \emptyset$ (closure taken with respect to ambient space $X$).
+Definition of discontinuity: $f:X\to Y$ is discontinuous at $x\in X$ if and only if  of $X$ and $\lim_{x\to p} f(q)$ either does not exist or $\neq f(x)$.
+Definition of monotonic functions: A function $f:(a,b)\to \mathbb{R}$ is monotone increasing if $\forall x>y$, we have $f(x) \geq f(y)$.
+Theorem: If $f$ is monotone, then $f(x)$ only has discontinuity of the first kind/simple discontinuity.
+Theorem: If $f$ is monotone, then there are at most countably many discontinuities.
+Definition of pointwise convergence of sequences of sequences:
+Definition of uniform convergenece of sequences of sequences:
+Definition of pointwise convergence of sequence of functions:
+Definition of uniform convergence:
+Given a sequence of functions $(f_n): X\to Y$, is said to converge uniformly to $f:X \to Y$, if $\forall \epsilon >0$, we have there exists $N>0$ such that $\forall n>N, \forall x\isin X$, we have $\lvert f_n(x) - f(x) \rvert <\epsilon$.
+Theorem: Suppose $f_n: X\to \mathbb{R}$ satisfies that $\forall \epsilon >0,\exists N>0$ such that $\forall x\in X, \lvert f_n(x) - f_m(x) \rvert < \epsilon$, then $f_n$ converges uniformly.
+Theorem: Suppose $f_n \to f$ uniformly on set $E$ in a metric space.  of $E$, and suppose that $\lim_{t\to x} f_n(t) = A_n$. Then $\{ A_n\}$ converges and $\lim_{t\to x} f(t) = \lim_{n\to\infty} A_n$. In conclusion, $\lim_{t\to x} \lim_{n\to\infty} f_n(t) = \lim_{n\to\infty} \lim_{t\to x} f_n(t)$.
+** After Midterm 2 **
+Definition of derivatives: Let $f: [a, b] \to \mathbb{R}$ be a real valued function. Define $\forall x \in [a, b]$
+$f'(x) = \lim_{t \to x} \frac{f(t) - f(x)}{t - x}$
+This limit may not exist for all points. If $f'(x)$ exists, we say $f$ is differentiable at $x$.
+Proposition: if $f:[a, b] \to \mathbb{R}$ is differentiable at $x_0 \in [a, b]$ then $f$ is continuous at $x_0$. $\lim_{x \to x_0} f(x) = f(x_0)$
+Theorem: Let $f, g: [a, b] \to \mathbb{R}$. Assume that $f, g$ are differentiable at point $x_0 \in [a, b]$, then
+$\forall c \in \mathbb{R}$, $(c \dot f)')x_0) = c \dot f'(x_0)$
+$(f + g)'（x_0） = f'(x_0) + g'(x_0)$
+$(fg)'(x_0) = f'(x_0) \cdot g(x_0) + f(x_0) \cdot g'(x_0)$
+if $g(x_0) \neq 0$, then $(f/g)'(x_0) = \frac{f'g - fg'){g^2(x_0)}$
+Mean value theorem: Say $f: [a, b] \to \mathbb{R}$. We say $f$ has a local minimum at point $p \in [a, b]$. If there exists $\delta > 0$ and $\forall x \in [a, b] \cap B_S(p), we have $f(x) \leq f(p).
+Proposition: Let $f: [a, b] \to \mathbb{R}$ If $f$ has local max at $p \in (a, b)$, and if $f'(p)$ exists, then $f'(p)=0$.
+Rolle theorem: Suppose $f: [a, b] \to \mathbb{R}$ is a continuous function and $f$ is differentiable in $(a, b)$. If $f(a) = f(b)$, then there is some $c \in (a, b)$ such that $f'(c) = 0$.
+Generalized mean value theorem: Let $f, g: [a, b] \to \mathbb{R}$ be a continuous function, differentiable on $(a, b)$. Then there exists $c \in (a, b)$, such that
+$(f(a) - f(b)) \cdot g'(c) = [g(a) - g(b)] \cdot f'(c)$
+Theorem: Let $f: [a, b] \to \mathbb{R}$ be continous, and differentiable over $(a, b)$. Then there exists $c \in (a, b)$, such that
+$[f(b) - f(a)] = (b - a) \cdot f'(c)$
+Corollary: Suppose $f:[a, b] \to \mathbb{R}$ continous $f'(x)$ exists for $x \in (a, b)$, and $|f'(x)| \leq M$ for some constant $M$, then $f$ is uniformly continous,
+Corollary: Let $f:[a, b] \to \mathbb{R}$ continuous, and differentiable over $(a, b)$. If $f'(x) \geq 0 $ for all $x \in (a, b)$, then $f$ is monotone increasing.
+If $f'(x) > 0$ for all $x \in (a, b)$, then $f$ is strictly increasing.
+Theorem: Assume $f, g: (a, b) \to \mathbb{R}$ differentiable, $g(x) \neq 0$ over $(a, b)$. If either of the following condition is true
+(1) $\lim_{x \to a}f(x) = 0, \lim_{x \to a}g(x) = 0
+(2) $\lim_{x \to a}g(x) = \infty$
+And if $\lim_{x \to a} \frac{f'(x)}{g'(x)} = A \in \mathbb{R} \cup {\infty, -\infty}$
+then $\lim_{x \to a} \frac{f(x)}{g(x)} = A$
+Higher derivatives: If $f'(x)$ is differentiable at $x_0$, then we define $f''(x_0) = (f')'(x_0)$. Similarly, if the $(n - 1)$-th derivative exists, n-th derivative.
+Definition of smooth: $f(x)$ is a smooth function on $(a, b)$ if $\forall x \in (a, b)$, if $\forall x \in (a, b)$, any order derivative exists.
+Taylor theorem: Suppose $f$ is a real function on $[a,b]$, $n$ is a positive integer, $f^{(n-1)}$ is continuous on $[a,b]$, $f^{(n)}(t)$ exists for every $t\isin (a,b)$. Let $\alpha, \beta$ be distinct points of $[a,b]$, and define $P(t) = \sum_{k=0}^{n-1} \frac{f^{(k)}(\alpha)}{k!} (t-\alpha)^k$. Then there exists a point $x$ between $\alpha$ and $\beta$ such that $f(\beta) = P(\beta) + \frac{f^{(n)}(x)}{n!}(\beta - \alpha)^n$.
+Taylor series for a smooth function If $f$ is a smooth function on $(a,b)$, and $\alpha \isin (a,b)$, we can form the Taylor Series: \\ $P_{\alpha}(x)=\sum_{k=0}^{\infty} \frac{f^{(k)}(\alpha)}{k!} (x-\alpha)^k$.
+Definition of Nth order taylor expansion:
+$P_{x_o,N}(x) = \sum_{n=0}^{N} f^{n)}(x_o) * \frac{1}{n!} (x-x_o)^n$.
+Definition of Partition: Let $[a,b]\subset \Reals$ be a closed interval. A partition $P$ of $[a,b]$ is finite set of number in $[a,b]$: $a=x_0 \leq x_1 \leq ... \leq x_n=b$. Define $\Delta x_i=x_i-x_{i-1}$.

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