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Math Help - Analysis proof continuous from left PLEASE HELP :)

  1. #1
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    Exclamation Analysis proof continuous from left PLEASE HELP :)

    I've been struggling on this one for hours so if someone could help me I would really really appreciate it!

    Let f be a bounded function on [a,b]. Show that the function defined by m(x) = inf{f(w):w in [a,x)} is continuous from the left on (a,b).

    I'm really looking forward to seeing your thoughts on this one! Thank you so much!
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  2. #2
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    Re: Analysis proof continuous from left PLEASE HELP :)

    Quote Originally Posted by Marissa043 View Post
    Let f be a bounded function on [a,b]. Show that the function defined by m(x) = inf{f(w):w in [a,x)} is continuous from the left on (a,b).
    This is an easy to prove question. But it depends upon set theory.
    If the function f is bounded and (a,y)\subset (a,x)\subset [a,b] then \inf\{f(t):t\in (a,x)\}\le\inf\{f(t):t\in (a,y)\}.

    That is, for bounded sets G \subseteq H \Rightarrow \inf (H) \leqslant \inf (G). Thus as defined the function m is non-increasing.

    So if s \in (x - \delta  ,x) then [a,s)\subset [a,x) and m(s)\ge m(x).

    Can you finish?
    Last edited by Plato; May 6th 2013 at 03:20 AM.
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    Re: Analysis proof continuous from left PLEASE HELP :)

    Well I was trying to use the definition: f is continuous from the left at x0 if f(x0)=limx->x0- f(x). But I'm not really sure what to do from your last step to get there. Can you lead me in the right direction please?
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    Re: Analysis proof continuous from left PLEASE HELP :)

    Quote Originally Posted by Marissa043 View Post
    Well I was trying to use the definition: f is continuous from the left at x0 if f(x0)=limx->x0- f(x). But I'm not really sure what to do from your last step to get there. Can you lead me in the right direction please?
    The key here is that m is a non-increasing function.
    We know that \forall x\in(a,b],~m(x)=\inf\{f(t):t\in[a,x)\} exists.

    If \varepsilon  > 0, so \exists f(t)\text{ such that }m(x)<f(t)<m(x)+\varepsilon where t\in[a,x).

    Let \delta=x-t>0 if y\in(x-\delta,x) we see that t=x-\delta<y<x.

    We know that m(y)\le m(t)<m(x)+\varepsilon  .

    Now can you finish?
    Thanks from Marissa043
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