Let D be a bounded subset of ℝ and f: D →ℝ be a bounded function. If the boundary ∂D of D is of

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Let D be a bounded subset of ℝ and f: D →ℝ be a bounded...

Let $D$ be a bounded subset of $\mathbb{R}$ and $f: D \rightarrow \mathbb{R}$ be a bounded function. If the boundary $\partial D$ of $D$ is of content zero and if the set of discontinuities of $f$ is also of content zero, then show that $f$ is integrable. In particular, if $D$ is of content zero, then show that $f$ is integrable and its Riemann integral is equal to zero. (Compare Remark 6.8.)

Key Concepts

Bounded Functions and Domains

A function defined on a bounded subset of the real numbers is considered bounded if its values remain within some fixed finite limits. A bounded domain means that the set over which the function is defined lies entirely within a finite interval. These conditions are fundamental in Riemann integration since the existence of a finite upper bound on both the function and the domain is required for defining the Riemann integral.

Content Zero (Jordan Measure Zero)

A set is said to be of content zero if, for every ? > 0, it can be covered by a finite collection of intervals whose total length is less than ?. This notion aligns with the idea of having negligible size, and it is crucial in Riemann integration as it implies that sets of content zero do not contribute to the value of the integral.

Riemann Integrability Criterion

A bounded function is Riemann integrable on a bounded domain if the set of its discontinuities has content (or measure) zero. This essential criterion ensures that the 'problematic' points where the function may not be continuous do not significantly affect the computation of the integral, allowing for the approximation of the integral by Riemann sums.

Integration Over Domains of Content Zero

When the entire domain of integration is of content zero, any bounded function defined on this set is Riemann integrable, and its integral equals zero. This happens because the domain can be covered by a collection of intervals with arbitrarily small total length, rendering the Riemann sums, and hence the integral, negligibly small.

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