Give an example of a function that is integrable on the interval [-1,1], but not continuous on [-1,1]. Explain.

consider the function f(x)= 0 -1<x<0 and = 3 for 0<x<1

Integrated:
F= int f(x)dx over -1 to 0 + int f(x)dx from - to 1
= 0 + 7

am I missing something here?

One example of such a function is the Dirichlet function, denoted as D(x).

To understand why the Dirichlet function is integrable on the interval [-1,1] but not continuous on [-1,1], it is important to understand the definitions of integrability and continuity.

A function is said to be integrable on a closed interval if it can be approximated by a sequence of simple functions whose integrals converge to a finite value. In other words, if the function can be divided into smaller pieces (subintervals) and the sum of the areas under these pieces approaches a finite value, then the function is integrable on that interval.

On the other hand, a function is said to be continuous on an interval if there are no abrupt jumps or breaks in its graph within that interval. In other words, if you can draw the graph without lifting your pen, the function is continuous.

The Dirichlet function, D(x), is defined as follows:

D(x) = 1, if x is rational,
D(x) = 0, if x is irrational.

Now, let's see why D(x) is integrable on the interval [-1,1] but not continuous on [-1,1].

To show the integrability, we can approximate D(x) by a sequence of simple functions. One approach is to define a step function, S(x), as follows:

S(x) = 1, if x is rational, and
S(x) = 0, if x is irrational.

Since S(x) only takes the values of 0 and 1, it is a simple function. By constructing a sequence of such step functions that approach D(x) by refining the divisions of intervals, we can show that D(x) is integrable on the interval [-1,1].

On the other hand, D(x) is not continuous on [-1,1] because it has abrupt jumps between rational and irrational numbers. As you move along the x-axis, the function changes from 1 to 0 and vice versa suddenly whenever the number switches between being rational and irrational.

Hence, the Dirichlet function serves as an example of a function that is integrable on the interval [-1,1] but not continuous on that interval.