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

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

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: math.stackexchange.com/questions/865560/… – Eric Auld Jul 13 '14 at 0:04