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\title[Test 2, 18.100B/C]
{Test 2 for Section 1 of 18.100B/C\\
To be submitted by 5PM Tuesday November 23.}

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NAME:\underline{\hspace{250pt}}
\vspace{40pt}

This is a take-home test. You should not talk to anyone else about the
questions or consult written, on-line or other material including your own
notes. I do not want time to be a major concern here so you can have up to
two hours in a single block after first reading the questions to complete
your answers. If for some reason you depart from these conditions you
should fully describe the circumstances under which you took the test.

The completed test can be dropped off in 2-108 or submitted electronically
to rbm at math dot mit dot edu.

\begin{enumerate}
\item If $f:X\longrightarrow Y$ is a continuous map between metric spaces
  and $K\subset X$ is compact, prove that $f(K)\subset Y$ is compact using
  the definition of compactness through open covers.
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NAME:\underline{\hspace{250pt}}
\vspace{40pt}
\item Suppose $f:[0,1]\longrightarrow \mathbb{R}$ satisfies
$$
|f(x)-f(y)|\le|x-y|^{\frac32}\ \forall\ x,y\in[0,1].
$$
Explain why it follows that $f(0)=f(1).$ 
\newpage
NAME:\underline{\hspace{250pt}}
\vspace{40pt}
\item If $\alpha :[0,1]\longrightarrow \mathbb{R}$ is given by
$$
\alpha(x)=\begin{cases}x-1&0\le x<\frac12\\
x+1&\frac12\le x\le 1
\end{cases}
$$
and $f=2x,$ explain why the Riemann-Stieltjes integral $\int_0^1fd\alpha$
exists and compute its value, justifying your arguments carefully.
\end{enumerate}

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