Exercise 8.1.4

Question

\begin{enumerate}[label=(\alph*)] 
\item Show that if $f$ is continuous, then it is possible to pick tags $\{c_k\}^n_{k=1}$ so that
$$R(f,P) = U(f,P)$$
Similarly, there are tags for which $R(f,P) = L(f,P)$ as well.
\item If $f$ is not continuous, it may not be possible to find tags for which $R(f,P) = U(f,P)$. Show, however, that given an arbitrary $\epsilon > 0$, it is possible to pick tags for $P$ so that
$$U(f,P) - R(f,P) < \epsilon$$
 \end{enumerate}

Ulisse Mini · Accepted Answer

\begin{enumerate}[label=(\alph*)] 
\item Each subinterval is closed, and since $f$ is continuous, the image of each subinterval under $f$ (the set of points which $f$ maps the subinterval to) is also closed, and thus contains its supremum and infimum; this allows us to pick tags so that $R(f,P) = U(f,P)$.
\item We can pick tags so that
$$M_k - f(c_k) < \frac{\epsilon}{b-a}$$
 \end{enumerate}

Exercise 8.1.4

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Exercise 8.1.4

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