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\textbf{Exercises on Bounded Variation}
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\noindent
Exercise 1.  Calculate $V_0^x f$ for each of the following functions $f$:

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(a) $f(x) = (x-1)^2$;

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(b) $f(x) = x(x-1)(x-2)$;

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(c) (d) $g(x) = \left \{
\begin{array}{cccc}
2x & \mbox{if } x \le 1  \\
3x & \mbox{if } 1 < x  . \\
\end{array} \right . $

\

\noindent
Exercise 2.  For each of exercises a - d below, calculate $\int_0^2 f dg$ for (i) $f=5x$ and for (ii) $f=x^2$ where $g$ is given below; then verify that your calculation is correct from the definitions:

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(a) $g=x^2$ ;

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(b) $g(x) = \left \{
\begin{array}{cccc}
0 & \mbox{if } x \le 1  \\
1 & \mbox{if } 1 < x  ; \\
\end{array} \right . $

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(c) $g(x) = \left \{
\begin{array}{cccc}
0 & \mbox{if } x < 1  \\
1 & \mbox{if } 1 \le x  ; \\
\end{array} \right . $

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(d) $g(x) = \left \{
\begin{array}{cccc}
2x & \mbox{if } x \le 1  \\
3x & \mbox{if } 1 < x  . \\
\end{array} \right . $

\

\noindent
Exercise 3a.  Determine if the following function is (a) R-integrable over $[0,1]$, (b) of bounded variation over $[0,1]$:

 $$f(x) = \left \{
\begin{array}{cl}
1 & \mbox{if } x = \frac 1n \mbox{ for } n \in \mathbb{Z}^+  \\
0 &  \mbox{elsewhere. } \\
\end{array} \right . $$

\noindent
Exercise 3b.  Repeat exercise 3a with the following function:

 $$f(x) = \left \{
\begin{array}{cl}
\sin(\frac 1x) & \mbox{if }  x>0  \\
0 &  \mbox{if }  x \le 0  \\
\end{array} \right . $$


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