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 10.2.3Let
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: