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\title{MTH436 - Homework 6}
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\begin{document}
 \maketitle
 
 Solutions to the questions in Section B should be submitted by the start of class on 4/4/18.
 
 
  
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 \section{Warm-up Questions}
 
 \begin{que}
  Suppose $f \colon [a,b] \to \R$ has the property that $f \in \Rint[c,b]$ for all $c \in (a,b)$. Is it true that $f \in \Rint[a,b]$? What if $f$ is bounded?
 \end{que}

 \begin{que}
  Let $f \colon [0,1] \to \R$ be defined by
  \[
   f(x) = \begin{cases}
           \sin \left( \frac{1}{x} \right) \quad &\text{if $x \in (0,1]$},\\
           0 \quad &\text{if $x = 0$.}
          \end{cases}
  \]
Is $f \in \Rint[0,1]$? Justify your answer.
 \end{que}

 
 \begin{que}
  Give an example of  $f \in \Rint[0,1]$ for which there is no $c \in [a,b]$ such that
  \[
   \int_a^b f = f(c)(b-a)
  \]
 \end{que}

 \begin{que}
  Let $a > 0$ and suppose $f \in \Rint[-a,a]$.
  \begin{enumerate}
   \item Show that if $f$ is even then $\int_{-a}^a f = 2 \int_0^a f$.
   \item Show that if $f$ is odd then $\int_{-a}^a f = 0$.
  \end{enumerate}

 \end{que}

 \begin{que}
  Let $f \colon [a,b] \to \R$ be continuous and $f(x) \geq 0$ for all $x \in [a,b]$. Show that if $\int_a^b f = 0$ then $f(x) = 0$ for all $x \in [a,b]$.
 \end{que}

 
 
 
 
 
 
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 \section{Submitted Questions}
 
 \begin{que}\mbox{}
 \begin{enumerate}
  \item Let $f \colon [a,b] \to \R$ be continuous. Prove that there exists $c \in [a,b]$ such that
  \[
   \int_a^b f = f(c)(b-a).
  \]
  \item Let $f,g \colon [a,b] \to \R$ be continuous and $g(x) > 0$ for all $x \in [a,b]$. Prove that there exists $c \in [a,b]$ such that
  \[
   \int_a^b fg = f(c) \int_a^b g.
  \]

  \end{enumerate}
 \end{que}

 \begin{que}
  Suppose $f \colon [-a^2,a^2] \to \R$ is continuous. Prove that
  \[
   \int_{{-}a}^a f\left(x^2\right) \, \mathrm{d}x = 2 \int_0^a f\left(x^2\right) \, \mathrm{d}x.
  \]
 \end{que}

 
 
 
 
 
 
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 \section{Challenge Questions}
 
 \begin{que}
  Let $f \colon [a,b] \to \R$ be continuous with $f(x) \geq 0$ for all $x \in [a,b]$. For $p \in \N$, define $\| f\|_p = \left( \int_a^b f^p \right)^{1/p}$ to be the $p$-norm of $f$. Prove that if we define $\|f\|_{\infty} = \sup \{ f(x) \mid x \in [a,b] \}$ then $\lim_{p \to \infty} \| f\|_p = \|f\|_{\infty}$.  
 \end{que}

 \begin{que}
  Using the notation from the previous questions, show that for $p=1,2$ and $\infty$, the function $d_p(f,g) = \|f-g\|_p$ defines a metric on the set $C([a,b])$ of continuous functions on $[a,b]$. 
 \end{que}

 
 
 \end{document}