Riemann Integral 2

Question 1

What are adv and dis adv of using Cauchy criterion

Answer 1

Adv of using Cauchy criterion is:
No need to compute upper and lower integral
Dis adv of using Cauchy criterion is:
No formula for integral b down to a

Question 2

If f is monotone increasing or decreasing, what else is it

Answer 2

If f is monotone increasing or decreasing, it is also integrable

Question 3

When is a function f uniformly continuous on I

Answer 3

Function f is uniformly continuous on I when:
For all epsilon, E delta for all c,x in I mod(x-c) < delta implies mod(f(x) - f(c)) < epsilon
In this, delta only depends on c whereas in continuous it depends on epsilon and c

Question 4

What is relationship between continuity and uniform continuity

Answer 4

Relationship between continuity and uniform continuity is:
Continuity DOES NOT imply uniform continuity in general
Uniform continuity implies continuity

Question 5

When does continuity imply uniform continuity

Answer 5

Continuity implies uniform continuity when:

F : [a,b] to R is continuous on [a,b] then f is uniformly continuous on [a,b]

Question 6

If f: [a,b] to R is continuous on [a,b], then what is f

Answer 6

If f: [a,b] to R is continuous on [a,b], then f is also integrable on [a,b]

Question 7

What is integral b down to a if b < a

Answer 7

Integral b down to a if b < a = - integral a down to b