# Pointwise convergence.

Definition 10.1.1   Let be a sequence of functions, all defined on the same domain . This sequence converges pointwise towards a function , defined on too if, for any , the sequence converges towards , i.e.

Note that depends not only on , but on the point too.

Example 10.1.2

Definition 10.1.3   Let be a sequence of functions. The series of functionsConsider the sequence of functions defined on the interval [0,1] by .

For every , the following holds:

, Thus for , the sum of the series is given by

and at 1, the series is divergent (sum of a series with a non zero constant general term). is pointwise convergent if the sequence of partial sums is pointwise convergent.

Example 10.1.4

Noah Dana-Picard 2007-12-28