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\centerline{Mathematics 242 -- Principles of Analysis}
\centerline{Discussion 3: Integrability}
\centerline{November 21, 2005}
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\noindent
{\it Background}
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On Friday, we introduced the upper and lower sums for a bounded
function $f$ on $[a,b]$ relative to a partition $P$:
$$\eqalign{U(f,P) &= \sum_{i=1}^n M_i \Delta x_i\cr
           L(f,P) &= \sum_{i=1}^n m_i \Delta x_i\cr}$$
where $M_i = {\rm sup} \{f(x) : x \in [x_{i-1},x_i]\}$, and 
$m_i = {\rm inf} \{f(x) : x \in [x_{i-1},x_i]\}$.  

We showed that every lower sum is less than or equal to every 
upper sum:
$$L(f,P) \le U(f,P')$$
for all partitions $P,P'$ of $[a,b]$, and we 
introduced the terminology that $f$ is {\it integrable\/}
over $[a,b]$ if 
$${\rm sup} \{L(f,P): P\ {\rm all\ partitions\ of\ } [a,b]\} = 
  {\rm inf} \{U(f,P): P\ {\rm all\ partitions\ of\ } [a,b]\}.$$
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\item{$(\ast)$} Another way to say that $f$ is integrable is to say that 
for any $\epsilon > 0$, there exists some partition $P$ of $[a,b]$
such that
$$U(f,P) - L(f,P) < \epsilon.$$
Using this alternate form of the definition, let's consider some 
examples and properties of integrable functions.
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\noindent
{\it Discussion Questions}
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\item{A)}  By using the definition $(\ast)$ and considering regular partitions
of $[0,2]$, show that $f(x) = 1 - 6x^2$ is 
integrable over $[0,2]$ and compute $\int_0^2 1 - 6x^2\, dx$.
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\item{B)}  Is the function 
$$f(x) = \cases{1 & if $x$ is a rational number\cr
                0 & if $x$ is not a rational number\cr}$$
integrable over $[0,1]$?  Why or why not?  (Recall from the first part
of the course that every interval of length $> 0$, no matter how small,
contains both rational and irrational numbers.)
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\item{C)}  Here's a somewhat surprising result.  Suppose $f$ is increasing
on the whole interval $[a,b]$.  Show that $f$ must be integrable over
$[a,b]$ by considering the difference $U(f,P_n) - L(f,P_n)$ for
a regular partition $P_n$ of $[a,b]$ and letting $n \to \infty$.  (Hint:  
$\Delta x = {b - a\over n}$ can be factored out of every term in 
 $U(f,P_n) - L(f,P_n)$.  When you do that, what happens in the sum?)
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\item{D)}  If $c \in (a,b)$ and $f$ is integrable over 
$[a,c]$ and $[c,b]$, then show $f$ is integrable over 
$[a,b]$ also, and 
$$\int_a^b f(x)\, dx = \int_a^c f(x)\, dx + \int_c^b f(x)\, dx$$
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\noindent
{\it Assignment}
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Writeups due Monday, November 28, no later than 5:00pm.
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