Uniform convergence.

Definition 10.2.1   If in Def. 1.1, $ N_0$ is independent of the point $ x$ , the sequence $ (f_n)$ is uniformly convergent.

Similarly, we define a uniformly convergent series of functions.

Example 10.2.2  

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

Example 10.2.4   Let $ F_n(x)=x^n$ , for $ 0 \leq x \leq 1$ . All the functions of the sequence are polynomial functions, thus they are continuous on $ [0,1]$ .

The sequence $ (F_n)$ is pointwise convergent; the limit is the function $ F$ defined as follows:

\begin{displaymath}\begin{cases}F(x)=0, x\in [0,1) F(1)=1 \end{cases}.\end{displaymath}    

The graph of $ F$ is displayed in Figure 1.
Figure: non uniform convergence of $ (x \mapsto x^n)$ on the interval $ [0,1]$ .
\begin{figure}\centerline
\mbox{\epsfig{file=NonUniformCv.eps,height=5cm}}\end{figure}
It follows that the sequence of functions $ (F_n)$ is not uniformly convergent.

Noah Dana-Picard 2007-12-28