7.3. Measures

Example 7.3.1(c): Oddities of Riemann Integral

What is the difference between Riemann integrable functions and bounded continuous functions?
There's little difference. A (bounded) Riemann integrable function must be continuous except possibly at countably many points. Therefore bounded functions that are Riemann integrable and those that are continuous can differ at most at countably many points.

That's odd: if there's little difference between these concepts, why are there two different concepts? Isn't one superfluous?

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