Chapter 24 Uniform Convergence

Question 1

Define Pointwise Convergence

Answer 1

(fn) converges pointwise to a function f defined on S if lim (fn(x)) as x approaches infinity = f(x).

Question 2

Explain fn(x) = x² for x in [0,1]. Whether or not it is point wise and why?

Answer 2

This is point wise because f(x) = 0 for x in [0,1) and f(x) = 1 for x = 1.

Question 3

What is uniform convergence?

Answer 3

(fn) converges uniformly on S to a function f defined on S if for each ε > 0 there exists a number N such that |fn(x) - f(x)| < ε for all x in S and all n > N.
f(x) - ε < fn(x) < f(x) + ε