A Riemann integrable function is a type of function that satisfies certain criteria for being integrable, meaning that its area under the curve can be calculated using a Riemann sum. This type of function is commonly used in calculus and is important for understanding the behavior of continuous functions.
This can happen when a function has a discontinuity or a singularity at one or more points in its domain. The Riemann integral requires that the function be continuous over its entire domain, whereas the absolute value only cares about the magnitude of the function, not its continuity. So, while the original function may not be integrable, its absolute value may still be integrable because it does not have any jumps or breaks in its graph.
The integrability of the absolute value allows for a way to still calculate the area under the curve of the original function, even though it may not be Riemann integrable. This is useful in cases where the function may have a few isolated points of discontinuity, but is otherwise continuous and well-behaved.
Yes, it is possible for a function to be not Riemann integrable and also have a non-integrable absolute value. This would occur in cases where the function is not continuous over its entire domain and has multiple points of discontinuity or a singularity that cannot be cancelled out by taking the absolute value.
Yes, there are other types of integrals, such as the Lebesgue integral, that can be used for functions that are not Riemann integrable. These integrals have different criteria for determining integrability and can handle more complicated functions than the Riemann integral. However, the Riemann integral is still widely used and studied due to its simplicity and importance in calculus.