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\title{MATH3210 - Spring 2024 - Section 004}
\author{Homework 9}

\begin{document}

\maketitle

\begin{problem}[30 points]
Let $a < d_1 < d_2 <\dots < d_k < b$ be a finite list, $g_i  : [d_i,d_{i+1}] \to \R$ be a continuous function for every $i = 1,\dots,k$, and $f : [a,b] \to \R$ be the function defined by

\[ f(x) = g_i(x) \qquad \mbox{when }x \in [d_i,d_{i+1}) \]

and $f(b) = g_k(b)$. Show that $f$ is integrable. 
\end{problem}


\begin{problem}[30 points]
Let $u$ and $v$ be continuously differentiable functions on $[a,b]$, and $V$ be an antiderivative of $v$. Show that

\[ \int_a^b uv \, dx = u(b)V(b) - u(a)V(a) + \int_a^bVu' \, dx. \]

[{\it Hint}: Apply the fundamental theorems to the fuction $H(y) = \displaystyle\int_a^y uv \, dx$]
\end{problem}

\begin{problem}[40 points]
Let $f_n : (a,b) \to \R$ be a sequence of functions on an open interval $(a,b)$. For each, prove or find a counterexample:

\begin{enumerate}[(a)]
\item If each $f_n$ is bounded, and $f_n \to f$ pointwise, then $f$ is bounded.
\item If each $f_n$ is bounded, and $f_n \to f$ uniformly, then $f$ is bounded.
\end{enumerate}
\end{problem}


\end{document}


