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:

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

The graph of $F$ is displayed in Figure 1.

Figure: non uniform convergence of $(x \mapsto x^n)$ on the interval $[0,1]$ .

It follows that the sequence of functions $(F_n)$ is not uniformly convergent.