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\title{MTH436 - Homework 8}
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\begin{document}
 \maketitle
 
 Solutions to the questions in Section B should be submitted by the start of class on 4/18/18. Unless specified, you should prove results directly from the definition of the Darboux Integral, not by invoking its equivalence to the Riemann Integral.
 
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 \section{Warm-up Questions}
 
 \begin{que}
  Define $f \colon [-1,1] \to \R$, $f(x) = \left|x + \frac{1}{2}\right| $. Compute $L(f,\calP)$ and $U(f,\calP)$ for the following partitions $\calP$.
  \begin{enumerate}
   \item $\calP = (-1,1)$.
   \item $\calP = (-1,0,1)$.
   \item $\calP = ({-}1,\,{-1/2},\,0,\,1/4,\,3/4,\,1)$.
  \end{enumerate}

 \end{que}
 
 
 \begin{que}
  Let $f_1,f_2 \colon [a,b] \to \R$ be bounded.
  \begin{enumerate}
   \item Show that if $\calP \in \mathscr{P}([a,b])$ then $U(f_1+f_2,\calP) \leq U(f_1,\calP) + U(f_2,\calP)$. 
   \item Deduce that $U(f_1 + f_2) \leq U(f_1) + U(f_2)$.
   \item Give an example where $U(f_1 + f_2) < U(f_1) + U(f_2)$.
  \end{enumerate}

 \end{que}

 \begin{que}
  Suppose $f,g \colon [a,b] \to \R$ are Darboux Integrable. Show that $f+g$ is Darboux Integrable and that
  \[
   \int_a^b f + g = \int_a^b f + \int_a^b g.
  \]

 \end{que}

 
 \begin{que}
  Let $f \colon [a,b] \to \R$ be bounded.
  \begin{enumerate}
   \item Suppose $k > 0$. Prove that $kU(f) = U(kf)$ and $kL(f) = L(kf)$.
   \item State and prove an analogous result to the above in the case where $k < 0$.
  \end{enumerate}

 \end{que}

 
 
 
 
 
 
 
 
 
 
 
 
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 \section{Submitted Questions}
 
 \begin{que}
  Let $f \colon [a,b] \to \R$ be Darboux Integrable and suppose $g \colon [a,b] \to \R$ satisfies $f(x) = g(x)$ for all $x \in [a,b] - E$, where $E$ is a finite set. Show that $g$ is Darboux Integrable and $\int_a^b f = \int_a^b g$. 
 \end{que}
 
 
 \begin{que}
  Let $f \colon [a,b] \to \R$ be bounded.
  \begin{enumerate}
   \item Show that if $f$ is continuous then it is Darboux Integrable.
   \item Show that if $f$ is monotone then it is Darboux Integrable.
  \end{enumerate}

 \end{que}

 
 
 
 
 
 
 
 
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 \section{Challenge Questions}
 
 
 \begin{que}
  \emph{The Riemann-Stieltjes Integral.} Let $\alpha \colon [a,b] \to \R$ be an increasing function and $f \colon [a,b] \to \R$ bounded. Given a partition $\calP = (x_0,x_1,\dotsc,x_n) \in \mathscr{P}([a,b])$, define
  \begin{align*}
   &U(f,\calP,\alpha) = \sum_{k=1}^n M_k (\alpha(x_{i}) - \alpha(x_{i-1})), \\
   &U(f,\calP,\alpha) = \sum_{k=1}^n m_k (\alpha(x_{i}) - \alpha(x_{i-1})).
  \end{align*}
where $M_k$ and $m_k$ are defined as for the Darboux Integral (note that if $\alpha(x) = x$ then this just reduces to the usual Darboux Integral). Then define
\[
 L_\alpha(f) = \sup \{ L(f,\calP,\alpha) \mid \calP \in \mathscr{P}([a,b]) \} \quad  \text{ and } \quad U_\alpha(f) = \inf \{ L(f,\calP,\alpha) \mid \calP \in \mathscr{P}([a,b]) \}.
\]
We say $f$ is Riemann-Stieltjes Integrable (and write $f \in \Rint(\alpha)$) if $L_\alpha(f) = U_\alpha(f)$. We denote this integral by $\int_a^b f \, \mathrm{d}\alpha$. 
\begin{enumerate}
 \item Prove the analogue for the Integrability Criterion for this integral.
 \item Show that if $f$ is continuous or monotone on $[a,b]$ then $f \in \Rint(\alpha)$.
 \item  Let $a < c < b$ and suppose $\alpha(x) = 0$ for $a \leq x < c$ and $\alpha(x) = 1$ for $c \leq x \leq b$. Prove that
 \[
  \int_a^b f\, \mathrm{d}\alpha = f(c).
 \]
 \item Suppose $\alpha' \in \Rint([a,b])$. Prove that $f \in \Rint(\alpha)$ if and only if $f \alpha' \in \Rint([a,b])$. Furthermore
 \[
  \int_a^b f \, \mathrm{d}\alpha = \int_a^b f(x) \alpha'(x) \, \mathrm{d}x.
 \]
\end{enumerate}

 \end{que}

 
 
 
 
 
 
 
 
 
 
 \end{document}