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Math Help - Necessary and sufficient condition for punctual and uniform convergence

  1. #1
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    Necessary and sufficient condition for punctual and uniform convergence

    Hello.
    Let h: R\longrightarrow{R} and consider f_n(x)= n(h(x + \frac{1}{n})-h(x)) find necessary and sufficient condition for:
    a) (f_n) is punctual convergent.
    b) (f_n) is uniform convergent into bounded intervals.

    I appreciate every contribution. Thank you so much.
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  2. #2
    Super Member girdav's Avatar
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    Re: Necessary and sufficient condition for punctual and uniform convergence

    What do we assume on h? For example, if h is a discontinuous function such that h(x+y)=h(x)+h(y) for all x,y\in\mathbb R, then the function f_n is constant, with a constant which doesn't depend on n.
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    Re: Necessary and sufficient condition for punctual and uniform convergence

    Quote Originally Posted by girdav View Post
    What do we assume on h? For example, if h is a discontinuous function such that h(x+y)=h(x)+h(y) for all x,y\in\mathbb R, then the function f_n is constant, with a constant which doesn't depend on 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.
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  4. #4
    Super Member girdav's Avatar
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    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.
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