Is f(x) integrable on the interval -1 < x < 1?

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SUMMARY

The function f(x) defined as f(x) = 1 for -1 < x < 0 and f(x) = -1 for 0 < x < 1 is integrable on the interval -1 < x < 1. To prove this, one must show that the superior integral (supremum of Riemann sums) and the inferior integral (infimum of Riemann sums) are equal. The discontinuity at x = 0 does not affect the integrability, as the function is bounded and the set of discontinuities has measure zero.

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Let

f(x)=

{1 if -1 < x<0;
{-1 if 0 < x < 1.

Prove that f(x) is integrable on -1 < x < 1.
 
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Just show that the superior and inferior integrals are equal.
 

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