If the function

is bounded abd that

is a partition of its domain [a,b]. For each index

we have:

and

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Then for a function

defined by:

in this case, we suppose

a partition of its domain [0,1]. since the rationals and irrationals are dense in R, it follows that for each index

, if

&

are defined as above, then

and

.

Therefore, the collection of the lower Darboux sums consists of the single number 0,

and the upper collection of darboux sums consists of the single number 1, therefore:

note: the * sign at the bottom and top denotes the lower integral of f on [a,b] and the higher integral of f on [a,b] respectively.