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In summary, the mean value theorem for integration can be used to prove the statement that ##a\lt \int_0^1f(x)dx \lt b## if ##f(x)## is a continuous function with ##a=inf(f(x))\lt f(x) \lt b=sup(f(x))##. Alternatively, one can use the general version of the mean value theorem with two functions, or write the integral as a limit of Riemann sums.

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Mean value theorem for integration: https://en.wikipedia.org/wiki/Mean_value_theorem#Mean_value_theorems_for_integrationmathman said:

It requires a continuous function ##f(x)##, so one probably has to have a look on the proof. But if ##f## is continuous, we have ##\int_0^1f(x)dx = f(\xi)\cdot (1-0)## with a mean value ##a<f(\xi)<b##.

The general version has two functions: ##g## continuous, and ##f## integrable, and says ##\int_0^1f(x)g(x)dx=g(\xi)\int_0^1f(x)dx##. Maybe one can find an appropriate ##g## and apply this general version.

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For a function to be integrable on the interval [0,1], it means that the area under the curve of the function on the interval [0,1] can be calculated using a definite integral. This means that the function has a finite value and can be integrated using different mathematical techniques.

To prove that a function is integrable on the interval [0,1], you can use the Riemann integral definition, which states that a function is integrable if the upper and lower Riemann sums converge to the same value. You can also use other methods such as the Fundamental Theorem of Calculus or the Lebesgue integrability criteria.

No, not all functions can be proven to be integrable on the interval [0,1]. If a function is discontinuous or unbounded on the interval, it may not be integrable. However, some functions that may seem to be discontinuous or unbounded can still be proven to be integrable using different mathematical techniques.

Proving the integrable function property is important because it allows us to calculate the area under the curve of a function, which has many real-world applications in fields such as physics, engineering, and economics. It also helps us to understand the behavior of functions and their properties.

Yes, the integrable function property has many practical implications. For example, it allows us to calculate the work done by a force, the distance traveled by an object, or the total profit of a business over a certain time interval. It also helps us to solve optimization problems and make predictions about the behavior of systems.

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