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\begin{document}
\begin{center}
{\bf {\large{Some Function Problems}}}\\
Isabel Vogt\\
Last Edited: \today \\
\end{center}
Most of these problems were written for my students in Math 23a/b at Harvard in 2011/2012 and 2012/2013. They cover basic function theory, countability, differential calculus, and point-set topology.
\begin{enumerate}
\item {\bf T/F: }The function $q:\mathbb{R} \rightarrow \mathbb{R}$ given by
$$ q(x)= \left\{
\begin{array}{lr}
x & : x \in \mathbb{Q}\\
0 & : x \notin \mathbb{Q}
\end{array}
\right.$$
is nowhere continuous. \\
\item {\bf T/F:} The space of results of ``infinite coin flips" (i.e. vectors where the $i^{\text{th}}$ position is H/T depending on the result of the flip) is countably infinite. \\
\item Let $f$ be a function $f:D \mapsto C$.
\begin{enumerate}
\item We can think of the function as merely a set of ordered pairs $(x,y)$. What set must this be a subset of?
\item Let $D=C=\mathbb{Q}$. And let $f$ be any function $f:D \mapsto C$. Thinking of $f$ as a set, what is its cardinality? Prove your answer.
\item Let $D=C=\mathbb{Q}$. What is the cardinality of the set of functions $f:D \mapsto C$. Justify your answer (it might help to think of what we call a similar set).
\item Let $D$ be an uncountable set, $C=\{0\}$. How many different well-defined functions $f$ are there? What can you say about this/these function(s) (i.e. injectivity, surjectivity, bijectivity)?
\item Let $D=\phi$ and $C$ be a countable set . How many different well-defined functions $f$ are there? Justify your answer based upon the your answer to part (a). If such an $f$ exists is it injective? Surjective?
\end{enumerate}
\item {\bf T/F:} Given a continuous function $f\begin{pmatrix} x \\ y \end{pmatrix}$ with $f: \R^2 \to \R$, if the Jacobian matrix at $\bold{a} \in \R^2$ exists, then $f$ is differentiable at $\bold{a}$, and its derivative is given by $[Jf(\bold{a}]$.
\item {\bf T/F:} $f$ is continuous at $x_0$ if and only if $\forall \epsilon>0, \exists \delta>0$ such that $f(B_{\delta}(x_0)) \subset B_{\epsilon}(f(x_0))$.
\item {\bf T/F:} If $f:\mathbb{R}^2 \rightarrow \mathbb{R}^2$ is differentiable and $[Df(0)]$ is not invertible, then there is no differentiable function $g:\mathbb{R}^2 \rightarrow \mathbb{R}^2$ such that
$$(g \circ f)(\vec{x})=\vec{x}$$
\item {\bf T/F:} A set $F$ is closed implies that all points in $F$ are accumulation points.
\item {\bf T/F:} For $f$ and $g$ differentiable functions from $\R^2 \to \R$ $$\frac{\partial f}{\partial x}\frac{\partial g}{\partial y} = \frac{\partial f}{\partial y}\frac{\partial g}{\partial x}$$
\item {\bf T/F:} Every convergent subsequence of a sequence $\{a_i\}$ defined on a compact set $C$ converges to the same limit $a \in C $.
\item Suppose $g: \mathbb{R}^2 \rightarrow \mathbb{R}^3$ is a function such that the directional derivatives at $x_0$ along $v_1$, $v_2$, and $v_3$ are $e_1$, $e_2$, and $e_3$ respectively. What can we say about $g$? Prove your claim.
\item
\begin{enumerate}
\item Define the zero-locus in $\R^2$ of the polynomial $f(x,y)=y-x^2$ to be the set of points $G_f=\{x_0,y_0 \in \R | f(x_0,y_0)=0\}$. Is the set $G_f$ open or closed as a subset of $\R^2$?
\item It is one thing to claim that there is an open set, and another thing to actually construct one. Given a point $\bold{a}=\begin{pmatrix} 3 \\ 1 \end{pmatrix}$, we wish to construct the largest ball $B_r(\bold{a})$ around this point such all points in this ball are in the complement of $G_f$ - ie $B_r(\bold{a}) \subset \R^2 - C$. Find an equation that will determine the maximum $r$ of this ball in \emph{two} ways:
\begin{enumerate}
\item Finding an extreme value of the distance function between the point $\bold{a}$ and the parabola
\item Use the concept that the shortest line between a point and a line will be a perpendicular.
\end{enumerate}
\end{enumerate}
\item Let
$$f\begin{pmatrix} x \\y
\end{pmatrix} = \frac{x ^3y^2+x^4y}{x^4+y^4}.$$
$f$ is defined to be 0 at $\begin{pmatrix} 0 \\0
\end{pmatrix}$.
\begin{enumerate}
\item Using the definition of continuity, prove that $f$ is continuous at $\begin{pmatrix} 0
\\0\end{pmatrix}$.
\item Show that both partial derivatives are zero at
$\begin{pmatrix} 0
\\0\end{pmatrix}$ but that the function is not differentiable there.
Note: You must work from the definition of ``differentiable.''
\end{enumerate}
\item The topological definition of a continuous function is as follows:
$$f:D \subset \R \to C \subset \R \text{ is continuous } \Leftrightarrow \forall \text{ open sets } A \subset C, f^{-1}(A) \text{ is open in } D$$
Using the standard metric topology of open sets with which you are familiar, show that this definition is equivalent to the $\epsilon$-$\delta$ definition with which you are familiar (\emph{Hint: you may want to use the formulation given in a true/false question above}).
\item {\bf T/F:} Let $X$ and $Y$ be sets with $A_1,A_2 \subset X$. For all functions $f: X \to Y$
$$f(A_1 \cap A_2) = f(A_1) \cap f(A_2)$$
as subsets of $Y$. And for $B_1,B_2 \subset Y$,
$$f^{-1}(B_1 \cap B_2)=f^{-1}(B_1) \cap f^{-1}(B_2)$$
as subsets of $X$. \\
\item {\bf T/F:} Let $h\begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} x^2y \\ y^2z \end{pmatrix}$. By the implicit function theorem, at the point $\begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix}$ we have a continuously differentiable function $i$ locally expressing
$x$ and $z$ in terms of $y$ to give the locus of points whose image under $h$ is $\begin{pmatrix} 0 \\ 0 \end{pmatrix}$. \\
\item {\bf T/F:} If $a: \mathbb{N} \rightarrow \mathbb{R}$ is a sequence converging to 0, then for any $b: \mathbb{N} \rightarrow \mathbb{R}$, the sequence $(a_ib_i)$, for $i \in \mathbb{N}$ converges to 0\\
\item Give an example of a continuous function $f: \R \to \R$ such that the restriction of the domain of $f$ to $\Z \subset \R$ is a continuous map, or prove that one does not exist.
\item Give an example of a function $g: \R \to \R$ that has a local inverse at \emph{every} point, but for which no global inverse exists, or prove that one does not exist.
\item For $F_1,...,F_n...$ closed sets such that $F_i \subset \R^2$, let
$$F=\bigcap_i F_i$$
Give an example of a function $f: F \to \R$ which is differentiable, or prove that one does not exist.
\item Give an example of a sequence of vectors $\vec{v}_1,\vec{v}_2,... \in \R^n$ which is convergent but the length of each vector is not uniformly bounded by some $M$, or prove that one does not exist.
\item Give an example of a surjective map $g:\Q \to \R$ which injects into its image, or prove that one does not exist.
\item {\bf The Inverse and Implicit Function Theorems}
\begin{enumerate}
\item As I am sure you can imagine, the implicit function theorem has many important practical applications. Here is a somewhat whimsical one: \\
The school district's curriculum committee has accurately calculated that if they provide $x$ units of instruction in test-geared material and $y$ units of instruction in academic exploration they achieve test scores S and student happiness H given by the function
$$\begin{pmatrix} S \\ H \end{pmatrix} = f\begin{pmatrix} x \\ y \end{pmatrix}= \begin{pmatrix} x^3 + xy^{1/2} \\ y^2 -4x^{1/2} \end{pmatrix}$$
Currently the curriculum is dictating educational time such that $x=4, y=1, S=68, H=-7$. Fearing trouble with reelection due to student unhappiness, the committee wants to moderately increase $H$ to -2 and only decrease $S$ to 63. Use the inverse function theorem to determine appropriate values of $x$ and $y$ that will approximate their desired output. How accurate is your answer? Why? \\
\item Call $X$ the space of linear transformations$A: \R^2 \to \R^2$ with determinant $+1$. Show that locally you can parameterize the passive variable(s) of $X$ in terms of the active variable(s). \\
\item Call $Y$ the space of linear transformations $B:\R^3 \to \R^3$ such that $B\circ B^T=I$ ($B$ is known as a \emph{orthogonal matrix}). How would you apply the implicit function theorem to locally parameterize $Y$?
\begin{enumerate}
\item Using the following statement:
\vspace{5pt}
\emph{All matrices $A\in Y$ satisfy the equation $AA^T-I=0$, thus I have a function $F:\R^9 \to \R^9$ given by $F:A \to AA^T-I$ for which $Y$ is the locus.}
\vspace{5pt}
Is the matrix $[DF]$ ever onto?
\item How do you resolve this? Think about what you know about $AA^T$.
\end{enumerate}
\end{enumerate}
\end{enumerate}
\end{document}