Week 3

Question 1

Answer 1

Question 2

Answer 2

Question 3

Answer 3

Question 4

Answer 4

Question 5

Answer 5

Question 6

Answer 6

Question 7

Answer 7

Question 8

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

Answer 8

It is continuous at every point to every other point

Question 9

Answer 9

Question 10

Answer 10

Question 11

If f is lipschtiz continuous on Δ

Answer 11

Which then also implies that it is continuous on Δ

Question 12

Answer 12

Question 13

And example

Answer 13

Question 14

Answer 14

Question 15

Answer 15

Question 16

Answer 16

Question 17

Answer 17

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

Question 18

Answer 18

Question 19

Boundedness theorem

Answer 19

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

Question 20

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

Answer 20

Take δ_ε = ε/C in definition of uniform continuity