Exercise 7.2.7

Question

Let $f:[a, b] \rightarrow \mathbf{R}$ be increasing on the set $[a, b]$ (i.e., $f(x) \leq$ $f(y)$ whenever $x<y)$. Show that $f$ is integrable on $[a, b]$.

Ulisse Mini · Accepted Answer

Since $f$ is increasing, let $P_n$ be comprised of the $n+1$ evenly spaced points $\{x_0, \cdots, x_n\}$, with $\Delta x_k = x_k - x_{k-1} = \Delta_n$ (where $\Delta_n$ is a constant for a fixed $n$). Then
$$U(f,P) = \sum^n_{i=1} f(x_i) (x_i - x_{i-1}) = \Delta_n \sum^n_{i=1}f(x_i)$$
and
$$L(f,P) = \sum^{n-1}_{i=0} f(x_i) (x_{i+1} - x_i) = \Delta_n \sum^{n-1}_{i=0} f(x_i)$$
Therefore $U(f,P) - L(f,P) = \Delta_n(f(b) - f(a))$. For a given $\epsilon > 0$ if we set $\Delta_n < \epsilon / (f(b) - f(a))$ then we can use Theorem 7.2.8 to conclude $f$ is integrable.

Exercise 7.2.7

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Exercise 7.2.7

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