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Oscillation of a point x in [c, d] <N => oscillation of subintervals of partition <N

  1. Apr 14, 2012 #1
    Q: Suppose that the oscillation ω_f (x) of a function f is smaller than η at each point x of an interval [c,d]. Show that there must be a partition π of [c,d] s.t. the oscillation
    ωf([x_(k-1),x_k ])<η
    on each member of the partition.

    My solution (Rough sketch):

    This condition on x is local, so it must be true for a δ-neightborhood of x s.t. ωf(δ(x))<η. Now take a partition s.t. each subinterval [x_(k-1),x_k ]<δ. Thus, each subinterval is less than the δ from the δ-neightborhood of x, so then
    ωf([x_(k-1),x_k ])[itex]\leq[/itex]ωf(δ(x))<η. QED

    Is this logic too sloppy? If so, does anyone have any suggestions as to a more proper way to prove this?
     
  2. jcsd
  3. Apr 14, 2012 #2
    Re: Oscillation of a point x in [c, d] <N => oscillation of subintervals of partition

    "The oscillation ω_f (x) of a function f is smaller than η at each point x of an interval [c,d]" means [itex]\lim_{t \to 0^+} \omega_{[x-t,x+t]}f(x)=h<\eta\Longrightarrow \forall \epsilon>0\,\,\exists \delta>0\,\,s.t.[/itex]

    [itex]0<t<\delta\Longrightarrow \left|\omega_{[x-t, x+t]}f(x)-h\right|<\epsilon[/itex]
    .

    Can you take it from here?

    DonAntonio
     
  4. Apr 15, 2012 #3
    Re: Oscillation of a point x in [c, d] <N => oscillation of subintervals of partition

    That's a much nicer definition of what's going on. Now, I can simply just take a partition of [c, d] s.t. each subinterval of the partition is of equal length, specifically [x-t, x+t], which satisfies the definition of being less than η. Correct?
     
  5. Apr 15, 2012 #4
    Re: Oscillation of a point x in [c, d] <N => oscillation of subintervals of partition



    Well, no, since "t" depends on the particular [itex]x\in [c,d][/itex] are we working with -- and thus it' would have been wiser

    to denote it by [itex]t_x[/itex] --, but then you can argue as follows:

    Since clearly our original interval [itex] [c,d]\subset \cup_{x\in [c,d]}(x-t_x,x+t_x)[/itex] and it is a compact set in the

    real line, there exists a finite number of points....etc.

    DonAntonio
     
  6. Apr 15, 2012 #5
    Re: Oscillation of a point x in [c, d] <N => oscillation of subintervals of partition

    You lost me on the finite number of points part. I appologize, as my compactness knowledge is quite scarse.
     
  7. Apr 15, 2012 #6
    Re: Oscillation of a point x in [c, d] <N => oscillation of subintervals of partition



    Well, if you haven't yet studied compact sets I, for one, cannot help you. The continuation of my idea is:

    By compactness of [c,d] there exist a finite number of points [itex]x_1,...,x_n\,\,s.t.\,\,[c,d]\subset \cup_{i=1}^n (x_i-t_{x_i},x_i+t_{x_i})[/itex] , so now we can choose

    [itex]t:=\max_i\{t_{x_i}\}[/itex] and now yes: end the proof as you wanted before.

    DonAntonio
     
  8. Apr 15, 2012 #7
    Re: Oscillation of a point x in [c, d] <N => oscillation of subintervals of partition

    I've been briefly introduced to a few compactness arguments. I see what you mean now, thank you for your help.
     
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