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\begin{center} \textbf{Integrability, Lemmas and Hints} \end{center}

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Theorem 2.1b:  Suppose that $f: [a,b] \rightarrow \mathbb{R}$ is a function.  If the function $f$ is R-integrable, then it is bounded.

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Hint:  If $f$ is not bounded over $[a,b]$ then there exists a number $c \in [a,b]$ so that if $B$ is a number and $\delta > 0$ then there is a point $x \in (c-\delta, c+\delta)$ so that $|f(x)| > B$.

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Definition:
Let $S$ be a subdivision with $S: \{a=x_0< x_1 < x_2 < \ldots < x_{n-1} < x_n = b\}$ and let $f$ be a function.  For this particular subdivision $S$ and function $f$, define for each $i$,
\begin{eqnarray*}
y_i^m & = & \mbox{glb}\{ f(x_i^*) \ | \ x_i^* \in [x_{i-1}, x_i] \} \\
y_i^M & = & \mbox{lub}\{ f(x_i^*) \ | \ x_i^* \in [x_{i-1}, x_i] \}.
\end{eqnarray*}
Then for the function $f$ define
\begin{eqnarray*}
\mbox{Lower Sum}(S, f) & = & \sum_{i=1}^n y_i^m(x_i - x_{i-1}) \\
\mbox{Upper Sum}(S, f) & = & \sum_{i=1}^n y_i^M(x_i - x_{i-1}).
\end{eqnarray*}

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Lemma 2.2b. Let $S$ be a subdivision with $S: \{a=x_0< x_1 < x_2 < \ldots < x_{n-1} < x_n = b\}$ and let $f: [a,b] \rightarrow \mathbb{R}$.  Then $f$ is R-integrable over [a,b] if and only if it is true that if $\epsilon >0$ then there exists a number $\delta > 0$ so that if $S$ is a subdivision of $[a,b]$ with mesh less than $\delta$  then
\begin{eqnarray*}
\mbox{Upper Sum}(S, f) - \mbox{Lower Sum}(S, f) & < & \epsilon.
\end{eqnarray*}

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Hint: For each $n \in \mathbb{N}$ let $S_n$ be a subdivision of $[a,b]$ with $\mbox{mesh}(S_n) < \frac 1n$.  Argue that the sequential limit of $\{\mbox{Lower Sum}(S_n, f)\}_{n=1}^\infty$ is $\int_{[a,b]}f$.  (And observe that $\{\mbox{Upper Sum}(S_n, f)\}_{n=1}^\infty$ also converges to $\int_{[a,b]}f$.)

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Lemma 2.3b. If $f: [a,b] \rightarrow \mathbb{R}$ is continuous then $f$ satisfies the hypothesis of Lemma 2.2b.

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Hint: Use the uniform continuity theorem.

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Claim:   Lemmas 2.2b and 2.3b imply Theorem 2.1.

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Lemma 2.4b.  Suppose $f: [a,b] \rightarrow \mathbb{R}$ is continuous and $g: [b,c] \rightarrow \mathbb{R}$ is continuous and $h$ is defined as follows:
$$h(x) = \left \{
\begin{array}{cccc}
f(x) & \mbox{if } a\le x< b \\
 d & \mbox{if } x=b \\
g(x) & \mbox{if } b < x \le c. \\
\end{array} \right . $$
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