Darboux Sums, Upper/Lower integral

If the function is bounded and 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.

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So, if we have another function

then how does

and in this case???

Following the example above, isn't the collection of lower darboux sum supposed to be x here, and how is it shown that the upper integral of f is greater than (1/2)?