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\begin{document}
\noindent{\small
  \href{\myurl}{Dror Bar-Natan}:
  \href{\myurl/classes/}{Classes}:
  \href{\myurl/classes/\#1617}{2016-17}:
  \hfill\url{http://drorbn.net/?title=1617-257}
}

\begin{center}

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\noindent{\bf Do not turn this page until instructed.}

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\href{http://drorbn.net/?title=1617-257}{\large Math 257 Analysis II}

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{\LARGE Term Test 1}

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University of Toronto, November 1, 2016

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\noindent{\bf Solve 4 of the 5 problems on the other side of this page. }\\
Each problem is worth 25 points.\\You have an hour and fifty minutes to
write this test.

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\noindent{\bf Notes}
\begin{itemize}
\item No outside material other than stationary is allowed.
\item {\bf Neatness counts! Language counts!} The {\em ideal} written
solution to a problem looks like a page from a textbook; neat and clean
and consisting of complete and grammatical sentences. Definitely phrases like
``there exists'' or ``for every'' cannot be skipped. Lectures are mostly
made of spoken words, and so the blackboard part of proofs given
during lectures often omits or shortens key phrases. The ideal written
solution to a problem does not do that.
{\red
\item In red: post-exam additions/notes.

\item Do not write on this examination form! Only what you write in the examination booklets counts towards your grade.
}
\end{itemize}

\end{center}

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\begin{center}

\includegraphics[width=2.5in]{JellyCube.jpg}
\linebreak
{\small From \url{http://goronic.deviantart.com/art/Jelly-Cube-559481292}}

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{\bf Good Luck!}

\end{center}

\newpage

\noindent{\bf Solve 4 of the following 5 problems. } Each problem is worth
25 points. You have an hour and fifty minutes. {\bf Neatness counts! Language counts!}

\vfill\noindent{\bf Problem 1. } Let $A$ be a {\red non-empty} subset of a metric space $(X,d)$. Show that the distance function to $A$, defined by $d(x,A):=\inf_{y\in A}d(x,y)$, is a continuous function of $x$ and that $d(x,A)=0$ iff $x\in\bar{A}$.

\noindent{\small {\bf Tip. } ``Iff'' means ``if and only if'', and it always means that there are two things to prove.}

\noindent{\small {\bf Tip. } Don't start working! Read the whole exam first. You may wish to start with the questions that are easiest for you.}

\vfill\noindent{\bf Problem 2. } Let $(X,d)$ be a metric space.
\begin{enumerate}
\item Define ``$x_0$ is a limit point of a subset $A$ of $X$''.
\item Prove that if $X$ is compact and $A\subset X$ is infinite, then $A$ has at least one limit point.
\end{enumerate}

\vfill\noindent{\bf Problem 3. }
\begin{enumerate}
\item Define ``$f\colon\bbR^n\to\bbR^m$ is differentiable at a point $a\in\bbR^n$''.
\item Prove that if $f\colon\bbR^n\to\bbR^m$ is differentiable at a point $a\in\bbR^n$, then its differential is uniquely determined.
\item Show that the function $f\colon\bbR^2\to\bbR$ given by $f(x,y)=|x y|$ is differentiable at $(x,y)=(0,0)$.
\end{enumerate}

\noindent{\small {\bf Tip. } In math exams, ``show'' means ``prove''.}

\vfill\noindent{\bf Problem 4. }
\begin{enumerate}
\item State ``the chain rule'' about the differential of the composition of two functions $f\colon\bbR^n\to\bbR^m$ and $g\colon\bbR^m\to\bbR^p$.
\item By appropriately choosing functions $f\colon\bbR\to\bbR^2$ and $g\colon\bbR^2\to\bbR$, find the derivative of the function $h(x)=x^x$.
\end{enumerate}

\noindent{\small {\bf Tip. } In math exams, ``state'' means ``write the statement of, in full''.}

\noindent{\small {\bf Tip. } In math exams, ``find'' means ``find and explain how you found''.}

\vfill\noindent{\bf Problem 5. } Denote by $B_r$ the open ball of radius $r$ around $0\in\bbR^n$ (using, here and below, the Euclidean metric). It is given that a function $f\colon B_1\to\bbR^n$ satisfies $f(0)=0$ and
\[ \forall x,y\in B_1,\quad |f(y)-f(x)-(y-x)|\leq 0.1|y-x|.
\qquad\text{\red (original had ``$<$'' by mistake)} \]
{\red (Even better if I had kept it ``$<$'' but switched to ``$\forall x\neq y$'')}
\begin{enumerate}
\item Prove that $f$ is one to one.
\item Prove that $f$ is onto $B_{0.4}$. Namely, for every $z\in B_{0.4}$ there is some $x\in B_1$ such that $z=f(x)$.
\end{enumerate}

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\centerline{\bf Good Luck!}

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\end{document}