An increasing function is a mathematical concept that describes a function whose output values increase as the input values increase. In other words, as the input increases, the output also increases. Graphically, this is represented by a line that slopes upwards from left to right.
Riemann-integrability is a mathematical concept that describes the ability to calculate the area under a curve using a method called the Riemann sum. It is a way to find the exact value of a definite integral, which represents the net area between the curve and the x-axis.
In order to prove that an increasing function is Riemann-integrable everywhere, you must show that it meets the criteria for Riemann-integrability. This includes being bounded on a closed interval, having a finite number of discontinuities, and having a limit at each point in the interval. If the function meets all of these conditions, then it is Riemann-integrable everywhere.
Yes, it is possible for an increasing function to be non-Riemann-integrable. This can occur if the function is not bounded on a closed interval or has an infinite number of discontinuities within the interval. In these cases, the Riemann sum cannot be used to accurately calculate the area under the curve, and the function is considered non-Riemann-integrable.
Riemann-integrability is important because it allows us to accurately calculate the area under a curve, which has many real-world applications in fields such as physics, economics, and engineering. It also allows us to solve certain types of differential equations and make predictions about the behavior of a system over time. Additionally, Riemann-integrability is a fundamental concept in calculus and serves as the basis for more advanced integration techniques.