Give an example of a function that is integrable on the interval [-1,1], but not continuous on [-1,1].
To find a function that is integrable on the interval [-1,1] but not continuous on [-1,1], we can consider the function f(x) = 1/x.
This function is not defined at x = 0, so it is not continuous on the interval [-1,1]. However, it is integrable on this interval.
To see this, we can calculate the definite integral of f(x) from -1 to 1:
∫[from -1 to 1] 1/x dx = ln|x| [from -1 to 1]
Evaluating the integral, we get:
ln(1) - ln(-1)
Since ln(-1) is not defined, we use the principal value of the natural logarithm, which is defined as ln(x)=ln(|x|)+i×(π) if x<0.
So, ln(1) - ln(-1) equals 0 - (ln(1) + i(π)), which simplifies to -iπ.
Since the value of the definite integral is finite, we can conclude that the function f(x) = 1/x is integrable on the interval [-1,1] even though it is not continuous at x = 0.