Week 3 Flashcards

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5
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7
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8
Q

Let f be a function defined on an interval Δ which may be (un)bounded. Then f is uniformly continuous on Δ if

A

It is continuous at every point to every other point

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9
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10
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11
Q

If f is lipschtiz continuous on Δ

A

Which then also implies that it is continuous on Δ

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12
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13
Q

And example

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14
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15
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16
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17
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Disprove!
Find a single point (finite or infinite) where f is not uniformly continuous
Look for a point where f oscillates or grows

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19
Q

Boundedness theorem

A

If f is a continuous function on a closed bounded interval, then f is bounded

20
Q

How to prove that if f is Lipshcitz continuous on Δ then f is uniformly continuous on Δ

A

Take δ_ε = ε/C in definition of uniform continuity