3) Uniform Convergence and Functions

Question 1

What is Pointwise Convergence

Answer 1

Question 2

What is Uniform Convergence

Answer 2

Question 3

What is the relationship between pointwise and uniform convergence

Answer 3

Question 4

Define uniform convergence of a sequence of functions fn to a function f on a domain D using the concept of supremum

Answer 4

Question 5

Explain the relationship between uniform convergence of a sequence of continuous functions fn on an interval [a,b] and the continuity of their limit function f

Answer 5

Question 6

Assume that the functions fn : D ⊆ R → R are bounded on D for every n ≥ 1 and that fn → f uniformly on D. Prove that f : D → R is bounded on D

Answer 6

Question 7

What can be concluded about a function f if it is the limit of a converging sequence of functions in C[a,b]

Answer 7

Question 8

If fn : [a, b] → R is continuous for every n ∈ N and the pointwise limit of the sequence (fn)n≥1 is discontinuous on [a, b], what does this imply about the functions convergence

Answer 8

The convergence cannot be uniform

Question 9

What is the relationship between uniform convergence of a sequence of Riemann integrable functions and the Riemann integrability of their limit function

Answer 9