Question

Let f(x) = x              0 ≤ x ≤ 1/2        = 3 - x         1/2 <...

Let
f(x) = x              0 ≤ x ≤ 1/2
       = 3 - x         1/2 < x ≤ 1
Find a partition P of [0,1] such that U(f, P) - L(f, P) < 1/100

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Answer #1

Consider the regular partition   of .

Then      (Actually it is a telescoping series since f is increasing on [0,1/2])

Consider the regular partition   of .

Then (Actually it is a telescoping series since f is decreasing on [1/2,1])

Hence for the regular partition ,

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