Learn Prove how one-sided limits, oscillation and small interval covers determine integrability, then test the theorem on dense and uncountable exceptional sets.
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The discontinuity criterion will construct partitions with small upper-minus-lower gaps.
For every positive epsilon there must be a partition whose upper sum minus lower sum is below epsilon.
One-sided limits and local oscillation use suprema of entire sets of values.
No. Approximation to the least upper bound supplies the estimates even if no point attains it.
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Dense jumps, a Cantor set and cancellation
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Prove one-sided limits and countability of discontinuities from order, with precise endpoint conventions.
6 practice tasks.
Measure local variation by all nearby values, locate closed discontinuity thresholds, and budget countable interval covers.
6 practice tasks.
Connect compact oscillation thresholds to short interval covers and small Darboux gaps, proving both directions of the bounded-interval criterion.
6 practice tasks.
Construct an integrable function with dense discontinuities and identify the exact limits of null-set modification arguments.
6 practice tasks.