If in Def. 1.1
is independent of the point
, the sequence
is uniformly convergent.
Similarly, we define a uniformly convergent series of functions.
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
. All the functions of the sequence are polynomial functions, thus they are continuous on
is pointwise convergent; the limit is the function
defined as follows:
The graph of
is displayed in Figure 1
non uniform convergence of
on the interval
It follows that the sequence of functions
is not uniformly convergent.