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(1) Let f : [0, 1] → R be defined by
(
x − 1, x ∈ Q
,
1 − x, x ∈
6 Q
f (x) =
, 1 . Compute U (Pn , f ) and L (Pn , f ).
and Pn = 0, n1 , · · · , n−1
n
. We have
n
n
X
1X
1
k−1
n−1
1
1
U (Pn , f ) =
sup f (x) =
1−
=1−
= +
n k−1 ≤x≤ k
n k=1
n
2n
2 2n
k=1
n
n
and
n
X
1
L (Pn , f ) =
n
k=1
n
1X
inf f (x) =
k−1
k
n k=1
≤x≤ n
n
k−1
1
1
− 1 = − U (Pn , f ) = −
+
/
n
2 2n
Rb
(2) Suppose f is continuous on [a, b] with f (x) ≥ 0 for all x ∈ [a, b]. If a f (x) dx = 0,
show that f (x) = 0 for all x ∈ [a, b].
. Assu...