Properties of the Definite Integral

If $f$ is continuous or piecewise continuous on [a,b], then $f$ is integrable on [a,b] . Fortunately, the functions we will use in the rest of this book are all integrable as are the functions you are likely to need for applications. However, there are functions for which the limit of the Riemann sums does not exist, and those functions are not integrable.

A nonintegrable function:

The function $\mathrm{f}(x)=\left\{\begin{array}{l} 1 \text { if } \mathrm{x} \text { is a rational number } \\ 2 \text { if } \mathrm{x} \text { is an irrational number } \end{array}\right.$   (Fig 12) is not integrable on [0,3].

Proof: For any partition $P$, suppose that you, a very rational person, always select values of $\mathrm{c}_{\mathrm{k}}$ which are rational numbers. (Every subinterval contains rational numbers and irrational numbers, so you can always pick $\mathrm{c}_{\mathrm{k}}$ to be a rational number.)

Then $\mathrm{f}\left(\mathrm{c}_{\mathrm{k}}\right)=1$, and your Riemann sum, $YS$, is always

$\mathrm{YS}_{\mathrm{P}}=\sum_{\mathrm{k}=1}^{\mathrm{n}} \mathrm{f}\left(\mathrm{c}_{\mathrm{k}}\right) \Delta \mathrm{x}_{\mathrm{k}}=\sum_{\mathrm{k}=1}^{\mathbf{n}} 1 \Delta \mathrm{x}_{\mathrm{k}}=3$

Suppose your friend, however, always selects values of $\mathrm{c}_{\mathrm{k}}$ which are irrational numbers. Then $\mathrm{f}\left(\mathrm{c}_{\mathrm{k}}\right)=2 \text {, }$, and your friend's Riemann sum, $FS$, is always

$\mathrm{FS}_{\mathrm{P}}=\sum_{\mathrm{k}=1}^{\mathrm{n}} \mathrm{f}\left(\mathrm{c}_{\mathrm{k}}\right) \Delta \mathrm{x}_{\mathrm{k}}=\sum_{\mathrm{k}=1}^{\mathrm{n}} 2 \Delta \mathrm{x}_{\mathrm{k}}=2 \sum_{\mathrm{k}=1}^{\mathrm{n}} \Delta \mathrm{x}_{\mathrm{k}}=6$

Then $\lim _{\text {mesh } \rightarrow 0} \mathrm{YS}_{\mathrm{P}}=3 \text { and } \lim _{m e s h \rightarrow 0} \mathrm{FS}_{\mathrm{P}}=6 \text { so } \lim _{m e s h \rightarrow 0}\left(\sum_{k=1}^{n} f\left(c_{k}\right) \cdot \Delta x_{k}\right)$ does not exist, and this f is not integrable on [0,3] or on any other interval either.