Integration 1: Integratability

Question 1

Define a partition of [a,b]

Answer 1

Finite set of points x_0, x_1, …, x_n where

Question 2

Define upper and lower Riemann sums

Answer 2

Question 3

Lower Riemann integral

Answer 3

Question 4

Upper Riemann integral

Answer 4

Question 5

Denote that f is Riemann integrable on [a,b]

Answer 5

Question 6

P* is a refinement of P if

Answer 6

P € P*
That is every point of P is a point of P*

Question 7

Common refinement

Answer 7

Question 8

Relate refinements to Riemann sums

Answer 8

Question 9

Proof of refinement lemma

Answer 9

Question 10

Prove

Answer 10

Question 11

Riemann’s criterion of integrability

Answer 11

Question 12

Proof of Riemann’s criterion of integrability

Answer 12

Question 13

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Answer 13

f is bounded and monotonic on [a,b]

Question 14

Prove that any bounded, monotonic function on [a,b] is Riemann integrable on [a,b]

Answer 14

Question 15

Prove

Answer 15

(All continuities on [a,b] are Riemann integrable on [a,b])

Question 16

Piecewise integrable functions are integrable

Answer 16

Question 17

Prove composition of of Lipschitz functions

Answer 17

Question 18

If f is bounded on [a, b] and (below) for all sufficiently small ε>0 , then

Answer 18

Question 19

Prove that f is Riemann integrable on [a,b] when it has finitely many discontinuities

Answer 19

Question 20

General characterisation of Riemann integrable functions

Answer 20