Necessary and sufficient condition for punctual and uniform convergence

Hello.

Let and consider find necessary and sufficient condition for:

a) is punctual convergent.

b) is uniform convergent into bounded intervals.

I appreciate every contribution. Thank you so much.

Re: Necessary and sufficient condition for punctual and uniform convergence

What do we assume on ? For example, if h is a discontinuous function such that for all , then the function is constant, with a constant which doesn't depend on .

Re: Necessary and sufficient condition for punctual and uniform convergence

Quote:

Originally Posted by

**girdav** What do we assume on

? For example, if h is a discontinuous function such that

for all

, then the function

is constant, with a constant which doesn't depend on

.

We don't assume anything. We are trying to find conditions (for example, h has to be disctontinuos or continious, deribable or not...that kind of things so as to reach a) and b) ).

Thank you for answering.

Re: Necessary and sufficient condition for punctual and uniform convergence

Continuity is not necessary, since taking for h the characteristic function of the rational numbers, we find that f_n=0 for all n.