Necessary and sufficient condition for punctual and uniform convergence

Hello.

Let $\displaystyle h: R\longrightarrow{R}$ and consider $\displaystyle f_n(x)= n(h(x + \frac{1}{n})-h(x))$ find necessary and sufficient condition for:

a) $\displaystyle (f_n)$ is punctual convergent.

b) $\displaystyle (f_n)$ 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 $\displaystyle h$? For example, if h is a discontinuous function such that $\displaystyle h(x+y)=h(x)+h(y)$ for all $\displaystyle x,y\in\mathbb R$, then the function $\displaystyle f_n$ is constant, with a constant which doesn't depend on $\displaystyle n$.

Re: Necessary and sufficient condition for punctual and uniform convergence

Quote:

Originally Posted by

**girdav** What do we assume on $\displaystyle h$? For example, if h is a discontinuous function such that $\displaystyle h(x+y)=h(x)+h(y)$ for all $\displaystyle x,y\in\mathbb R$, then the function $\displaystyle f_n$ is constant, with a constant which doesn't depend on $\displaystyle n$.

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.