# Thread: Necessary and sufficient condition for punctual and uniform convergence

1. ## 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.

2. ## 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 $\displaystyle n$.

3. ## Re: Necessary and sufficient condition for punctual and uniform convergence

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) ).