The definition of integrability states that a function is integrable if it can be bounded between two finite numbers by finding the area under its curve over a given interval.
To prove that |f(x)-g(x)| is integrable on [a,b], you must show that the upper and lower sums of the function converge to the same value as the interval size approaches zero. This can be done by using the definition of integrability and evaluating the limit of the Riemann sums.
Proving integrability is important because it allows us to determine the area under a curve and find the definite integral of a function. This is useful in many fields, such as physics, engineering, and economics, where finding the total amount or quantity is necessary.
No, the Intermediate Value Theorem cannot be used to prove integrability. This theorem is used to show that a continuous function takes on all values between two points, but it does not prove integrability.
Absolute integrability means that the function f(x) is integrable on [a,b] if the absolute value of the function is integrable on [a,b]. Uniform integrability means that the function f(x) is integrable on [a,b] if for any interval [c,d] within [a,b], the function is also integrable on [c,d].