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Which integrable functions have the property that all lower sums are equal?

This is from Spivak's Calculus (a * problem). The question mentions the use of dense sets as a hint as well as the fact that if f is integrable on [a,b] then f must be continuous at many points in [a,b]

My question is, doesn't a constant function satisfy this? The lower sums (as well as the upper sums) will be all equal regardless of the partition... I guess I'm underthinking this. Can anybody help see where my thinking goes wrong and how I should proceed?

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Yes, constant functions satisfy it. But they need not be all the functions that satisfy it; Spivak is asking you to determine all functions that satisfy the condition, and for all we know at this stage, the constant functions may not completely exhaust the class. – Arturo Magidin Nov 16 '11 at 22:17
@Arturo Thank you – MathMathCookie Nov 16 '11 at 23:47
up vote 1 down vote accepted

Clearly, any function where the points which are mapped to the global minimum satisfies this condition. Also, any function which does not satisfy this condition cannot have all lower sums equal- pick a neighborhood without a point mapped to the global minimum as part of your subdivision.

Now, suppose that a function satisfying this is integrable. Then it should be discontinuous at countably many points. This should be enough (with some poking around) to narrow down a better characterization of what functions these are exactly.

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I disagree with your answer. If $\mathscr{C}$ is the Cantor set, then $\chi_{\mathscr{C}}$ is Riemann integrable and has all lower sums equal, though it is discontinuous at uncountably many points. See my question:… – Eric Auld Jul 13 '14 at 0:04

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