MA005: Calculus I

This important question was finally answered in the 1850s by Georg Riemann, a name that should be familiar by now. Riemann proved that a function must be badly discontinuous to not be integrable.

Every continuous function is integrable

If $f$ is continuous on the interval $[a,b]$

then $\lim _{m e s h \rightarrow 0}\left(\sum_{k=1}^{n} f\left(c_{k}\right) \cdot \Delta x_{k}\right)$ is always the same finite number, $\int_{a}^{b} f(x) d x$, so $f$ is integrable on $[a,b]$.

In fact, a function can even have any finite number of breaks and still be integrable.

Every bounded, piecewise continuous function is integrable.

If $f$ is defined and bounded $(-\mathrm{M} \leq \mathrm{f}(x) \leq \mathrm{M}$ for all $x$ in $[a,b]$ and continuous except at a finite number of points in $[a,b]$ ,

then $\lim _{m e s h \rightarrow 0}\left(\sum_{k=1}^{n} f\left(c_{k}\right) \Delta x_{k}\right)$ is always the same finite number, $\int_{a}^{b} f(x) d x$, so f is integrable on $[a,b]$.

The function $f$ in Fig. 9 is always between –3 and 3 (in fact, always between –1 and 3) so it is bounded , and it is continuous except at 2 and 3. As long as the values of f(2) and f(3) are finite numbers, their actual values will not effect the value of the definite integral, and

$\int_{0}^{5} \mathrm{f}(x) \mathrm{dx}=0+6+2=8$.

Practice 5: Evaluate $\int_{1.5}^{3.2} \mathrm{INT}(x) \mathrm{dx}$ . (Fig. 10)

Fig. 11 summarizes the relationships among differentiable, continuous, and integrable functions:

• Every differentiable function is continuous, but there are continuous functions which are not differentiable. (example: $|x|$ is continuous but not differentiable at $\x=0$.
• Every continuous function is integrable, but there are integrable functions which are not continuous. (example: the function in Fig. 9 is integrable on [$0, 5]$ but is not continuous at 2 and 3.)
• Finally, as shown in the optional part of this section, there are functions which are not integrable.