Read this section to learn about properties of definite integrals and how functions can be defined using definite integrals. Work through practice problems 1-5.
If is continuous or piecewise continuous on [a,b], then 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 (Fig 12) is not integrable on [0,3].
Proof: For any partition , suppose that you, a very rational person, always select values of which are rational numbers. (Every subinterval contains rational numbers and irrational numbers, so you can always pick to be a rational number.)
Then , and your Riemann sum, , is always
Suppose your friend, however, always selects values of which are irrational numbers. Then , and your friend's Riemann sum, , is always
Then does not exist, and this f is not integrable on [0,3] or on any other interval either.