Final exam solutions

# Ε if we choose ε f c 2 then we find that there is a

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) | < 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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