# Uniform convergence.

Definition 10.2.1   If in Def. 1.1, is independent of the point , the sequence is uniformly convergent.

Similarly, we define a uniformly convergent series of functions.

Example 10.2.2

Theorem 10.2.3   Let be a sequence of functions, all of them continuous over the same interval . If the sequence is uniformly convergent, then the limit is contonuous on .

Example 10.2.4   Let , for . All the functions of the sequence are polynomial functions, thus they are continuous on .

The sequence is pointwise convergent; the limit is the function defined as follows:

The graph of is displayed in Figure 1.
It follows that the sequence of functions is not uniformly convergent.

Noah Dana-Picard 2007-12-28