)

< ε.
If we choose
ε
=
f
(
c
)
/
2, then we find that there is a
δ >
0 such that

x

c

< δ
and
x
∈
[
a, b
] implies that
f
(
x
)
> f
(
c
)
/
2.
At least half of the interval
[
c

δ, c
+
δ
] lies inside [
a, b
], and on that interval we have
f
(
x
)
> f
(
c
)
/
2,
hence the integral of
f
(
x
) over the part of the interval [
c

δ, c
+
δ
] lying
inside [
a, b
] is greater than or equal to
δ
·
f
(
c
)
/
2
>
0, which contradicts the
assumption.
7
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 Spring '08
 Staff
 Continuous function, Natural number, Ks, lim sn

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