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{MA773}         \hfill {Russell Brown}\\ 
Final
\medskip	


Answer questions 1--3 and three of questions 4--8.    Indicate clearly
which three questions are to be graded.

\begin{enumerate}


\item (5 points) State the least upper bound property of the real
numbers.

\item (10 points) State the fundamental theorem of calculus.

\item (10 points) State the mean value theorem.


\item (25 points) Let $X$ be a compact metric space and $Y$ a metric
space. Let $f:X\rightarrow Y$, be  continuous, one-one  and
onto. Prove that $f^{-1}$ is continuous. 

\item (25 points) Recall that  a real-valued function on $\reals$ is
convex if for any three real numbers, $x<y<z$, we have 
$$
f(y) \leq f(x)\frac {z-y}{z-x} + f(z) \frac {y-x}{z-x}.
$$
Prove that if $f$ is twice differentiable and $f''(x) \geq0$, then $f$
is convex. 

\item  (25 points)
\begin{enumerate}
\item Let $f$ be a bounded real-valued function on $[0,1]$.  Define
the upper integral 
$$  \int_0^1 f \, dx.$$  

\item If $f$ and $g$ are two bounded,  real valued functions on $[0,1]$, show
that
$$
 \int_0^1 f+g \, dx \leq  \int_0^1 f \, dx +  \int _0^1 g
\, dx.  
$$
\item Is it possible that (b) is not an equality? Give a proof or example to
support your answer.
\end{enumerate}

\item (25 points)  Suppose that $f_n$ is a sequence of continuous real-valued
functions on the unit interval $[0,1]$ and that the sequence converges
uniformly to a function $f$. 
\begin{enumerate} 
\item Prove that the function $f$ is integrable. (Hint: This is easy.)
\item Show that
$$
\int _0^1 f \, dx = \lim_{n\rightarrow \infty}\int_0^1 f_n \, dx.
$$
\item Show that equality may fail to hold if we only assume that $
f_n(x) \rightarrow f(x)$ for each $x$, but do not require the
convergence to be uniform. 
\end{enumerate}

\item (25 points)
\begin{enumerate}
\item Suppose
 that $f_n$ is a sequence  of continuous real-valued  functions on
$ [0,1]$ which converge uniformly to a function $f$. Suppose that
$\{x_n\}$ is a sequence in $[0,1]$ which converges to $x_0$. Prove that
$$
\lim _{n\rightarrow \infty} f_n(x_n) = f(x_0).
$$

\item 
Show that the equality in part (a) may fail if the sequence $f_n$
does not converge uniformly. 

\end{enumerate}

\end{enumerate}

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