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AxiomOfChoice

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In summary, a function is Riemann integrable on a bounded interval only if its set of discontinuities has measure zero, according to the Riemann-Lebesgue lemma. However, this does not mean that the function is equal to a continuous function almost everywhere, as shown by the counterexample of f(x). However, for every e>0, there exists a set A of measure less than e and a continuous function g such that, outside of A, f and g are equal.

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AxiomOfChoice

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- #2

AxiomOfChoice

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[tex]

f(x) = \begin{cases}

1 & \text{ if } 0\leq x \leq 1/2\\

0 & \text{ if } 1/2 < x \leq 1

\end{cases}

[/tex]

Then [tex]f[/tex] is continuous almost everywhere, but it cannot be equal to a continuous function almost everywhere by an argument involving inverse images of open sets, etc. Bummer.

- #3

Hurkyl

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A Riemann integrable function is a function that can be integrated using the Riemann integral, which is a method of calculating the area under a curve. It is a fundamental concept in calculus and is defined as a function that is bounded and continuous on a closed interval [a, b].

A function is continuous except on a set of measure zero if it is continuous everywhere except for a set of points with no length, area, or volume. In other words, the function may have discontinuities at a finite number of points, but these points have no impact on the overall continuity of the function.

This condition is important because it ensures that the function is well-behaved and can be easily integrated using the Riemann integral. Without this condition, the function may have discontinuities that could make the integral difficult or impossible to calculate.

Yes, as long as the function is bounded and the set of discontinuities has no length, area, or volume. This is because the Riemann integral only considers the behavior of the function on a closed interval, and the discontinuities at a finite number of points will not affect the overall area under the curve.

To determine if a function is continuous except on a set of measure zero, we can use the Lebesgue criterion. This states that a function is continuous except on a set of measure zero if and only if the set of discontinuities has Lebesgue measure zero. This can be verified by checking the continuity of the function at each point and determining if the set of discontinuities has no length, area, or volume.

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