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{\bf Supplementary problems (assigned 11/3/03)}
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\begin{enumerate}
\item (Leftover differentiation problem) Suppose that $f:(a,b)\to\R$ is
a convex function that is differentiable at every $x\in (a,b)$.  Show
that $f'$ is continuous. (Hint: you can, of course, use the results of
previous homework problems about convexity; moreover, there is a theorem
in the book that makes this problem much easier---for once, it is not the
mean value theorem or the chain rule.)
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\item 
\vspace*{.2in} 
The function $1/t$ is continuous on $(0,\infty)$.  Therefore the function
$$
f(x) = \int_1^x \frac{dt}{t}.
$$
is well-defined for all $x\in (0,\infty)$.  Prove each of the following
about $f$.
\begin{itemize}
\item $f$ is differentiable at every point and strictly increasing.
\item $f(xy) = f(x) + f(y)$ for every $x,y\in (0,\infty)$.  It
helps here to think of $y$ as a constant so that both sides are functions of
$x$ only.
\item $f(x^t) = t f(x)$ for all $t\in\R, x\in (0,\infty)$.
Remember the problem from the first chapter in which $x^t$ was defined for
any real $t$---the idea was to do it first for $t\in\Z$, then for $t\in\Q$,
and then, using supremums, for $t\in\R$.
\item $f(0,\infty) = \R$.  In particular, there is a unique
number $d\in (1,\infty)$ such that $f(d) = 1.$
\item $f$ is an invertible function and that $f^{-1}(y) = d^y$
for all $y\in\R$.
\end{itemize}

\end{enumerate}
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