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\noindent \underline{MATH 4590 -- Study for final} \\

\noindent \underline{\textbf{Definitions}} \\
\begin{itemize}
\item Let $M$ be a set and let $d \colon M \times M \rightarrow [0,\infty)$ be a function. Define what it means for the ordered pair $(M,d)$ to be a metric space: It means that the function $d$ satisfies three axioms: \\
\begin{enumerate}
\item (Positive definite) $d(x,y) \geq 0$ and $d(x,y)=0$ if and only if $x=y$,
\item (Symmetry) for all $x, y \in M$, $d(x,y)=d(y,x)$, and
\item (Triangle inequality) for all $x,y,z \in M$, $d(x,y) \leq d(x,z) + d(z,y)$.
\end{enumerate}
\item Let $(M,d_1)$ and $(N,d_2)$ be metric spaces. Define what it means for $f \colon M \rightarrow N$ to be a continuous function: It means $\forall \epsilon > 0 \forall p \in M \exists \delta >0$ such that if $q \in M$ and $d_1(p,q)<\delta$, then $d_2(f(p),f(q))<\epsilon$. 
\item Let $(M,d)$ be a metric space, let $p \in M$, and let $(p_n)$ be a sequence in $M$.  Define what it means for $p_n$ to converge to $p$, i.e. $p_n \rightarrow p$: It means $\forall \epsilon > 0 \exists N \in \mathbb{N}$ such that if $n \geq N$, then $d(p_n,p)<\epsilon$. 
\item Let $(M,d)$ be a metric space and let $S \subset M$. Define what a limit of $S$ is: We say that $p \in M$ is a limit of $S$ provided there is a sequence $(p_n)$ in $S$ such that $p_n \rightarrow p$.
\item Let $(M,d)$ be a metric space and let $S \subset M$. Define what it means for $S$ to be a closed set: We say that $S$ is closed if it contains all of its limits.
\item Let $(M,d)$ be a metric space and let $S \subset M$. Define what it means for $S$ to be an open set: We say that $S$ is open if $\forall p \in S \exists r>0$ so that $\{q \in M \colon d(p,q)<r \} \subset S$. 
\item Define what $\displaystyle\lim_{t \rightarrow a} f(t)=L$ means: $\forall \epsilon >0 \exists \delta > 0$ such that$|t-a| < \delta$ implies $|f(t)-L|<\epsilon$.
\item Let $f \colon \mathbb{R} \rightarrow \mathbb{R}$. Define the derivative of $f$ at $x$: $f'(x) = \displaystyle\lim_{t \rightarrow x} \dfrac{f(t)-f(x)}{t-x}$
\item $P, T$ partition pair of $[a,b]$: $P=\{x_0,x_1,\ldots,x_n\} \subset [a,b]$ and $T=\{t_1,\ldots,t_n\} \subset [a,b]$ with $x_0 < t_1 < x_1 < t_2 < x_3 < \ldots < t_n < x_n$.
\item $f$ Riemann integrable on $[a,b]$ with integral $I$: $\forall \epsilon > 0 \exists \delta > 0$ such that if $P,T$ is a partition pair of $[a,b]$ with $\mathrm{mesh}(P)<\delta$, then $|R(f,P,T) - I|<\epsilon$
\item If $f_n \colon [a,b] \rightarrow \mathbb{R}$ is a sequence of functions, then $f_n \rightarrow f$ (pointwise convergence) means $\forall x \in [a,b] \forall\epsilon >0 \exists N \in \mathbb{N}$ so that if $n \geq N$, then $|f_n(x)-f(x)|<\epsilon$ 
\item If $f_n \colon [a,b] \rightarrow \mathbb{R}$ is a sequence of functions, then $f_n \rightrightarrows f$ (uniform convergence) means $\forall \epsilon>0 \exists N \in \mathbb{N}$ so that if $n \geq N$ and $x \in [a,b]$, then $|f_n(x)-f(x)|<\epsilon$.
\item Outer measure: if $X$ is a set, a function $m^* \colon \mathscr{P}(X) \rightarrow [0,\infty)$ is called an outer measure if $m^*(\emptyset)=0$, it is monotone meaning if $A, B \in \mathscr{P}(X)$ with $A \subseteq B$, then $m^*(A) \leq m^*(B)$, and it is countably subadditive meaning if $A_n \in \mathscr{P}(X)$ for $n=1,2,3,\ldots$, then $m^* \left( \displaystyle\bigcup_{n=1}^{\infty} A_n \right) \leq \displaystyle\sum_{k=1}^{\infty} m^*(A_n)$
\item Measure: if $X$ is a set, a function and $\mathcal{M}$ is a set of so-called ``measurable subsets" of $X$, then $m \colon \mathcal{M} \rightarrow [0,\infty)$ is called a measure if $m(\emptyset)=0$, it is monotone meaning if $A, B \in \mathcal{M}$ with $A \subseteq B$, then $m(A) \leq m(B)$, and it is countably additive meaning if $A_n \in \mathcal{M}$ for $n=1,2,3,\ldots$ are mutually disjoint, i.e. $A_i \cap A_j = \emptyset$ iff $i \neq j$, then $m \left( \displaystyle\bigcup_{n=1}^{\infty} A_n \right) = \displaystyle\sum_{k=1}^{\infty} m(A_n)$
\end{itemize}

\newpage

\underline{Topics from Exam 1} \\
\begin{enumerate}
\item proving a sequence of real numbers converges (problem 2)
\item proving a quadratic real-valued polynomial function is continuous (problem 3)
\item metric or not? (problem 4)
\item identify unit circles in various metric spaces (problem 5)
\item prove a function involving a function already known to be continuous is also continuous (problem 6)
\item (bonus): prove product of continuous functions is continuous
\end{enumerate}

\underline{Topics from Exam 2}
\begin{enumerate}
\item open set or closed set or both? (problem 2)
\item prove the limit of a polynomial is what it is (problem 3)
\item proving finite union or finite intersection of open/closed sets is open/closed (problem 4)
\item induction proofs involving calculus operations (problem 5)
\item taylor series problem (problem 6)
\item proof involving compactness (problem 7)
\item proof involving mean value theorem (problem 8)
\end{enumerate}

\underline{Topics from Exam 3}
\begin{enumerate}
\item proofs involving uniform convergence
\item finding pointwise limits 
\item deciding whether or not a pointwise converging sequence is also uniformly converging
\item deciding if changing the domain changes whether convergence is uniform or not
\item induction proof involving sums
\item proof involving properties of integrals
\item (bonus) refinement principle
\item (bonus) draw picture of uniform convergence as described
\item (bonus) proof involving zero sets
\end{enumerate}

\underline{Since Exam 3}
\begin{enumerate}
\item HW11
\begin{enumerate}
\item calculating Bernstein polynomials
\item Weierstrass approximation theorem stuff
\end{enumerate}
\item HW12
\begin{enumerate}
\item calculating Lebesgue measure of subsets of $\mathbb{R}$
\item drawing pictures of simple functions
\item integrating simple functions
\item $f^+$ and $f^-$
\end{enumerate}
\item Quizzes
\begin{enumerate}
\item set addition -- e.g. $[3,6]+5$ (23 April notes)
\item proving something is an outer measure (did cardinality in class...could be anything) (23 April notes)
\item calculating with ``addition modulo 1", $\oplus$ (25 April notes)
\item prove the $\sim$ relation defined by $x \sim y$ iff $x-y \in \mathbb{Q}$ is an equivalence relation (27 April notes)
\item stuff similar to HW12 (27 April notes)
\end{enumerate}
\end{enumerate}
\end{document}