ABSTRACT
Definition 8.1 A sequence of functions {fj}∞j=1 with domain S ⊆ R is said to converge pointwise to a limit function f on S if, for each x ∈ S, the sequence of numbers {fj(x)} converges to f(x). Example 8.2 Define fj(x) = x
j with domain S = {x : 0 ≤ x ≤ 1}. If 0 ≤ x < 1 then fj(x) → 0. However, fj(1) → 1. Therefore the sequence fj converges pointwise to the function
f(x) =
{ 0 if 0 ≤ x < 1 1 if x = 1
See Figure 8.1. We see that, even though the fj are each continuous, the limit function f is not.
