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On equicontinuity

  1. Nov 29, 2007 #1
    say a function f is continuous on the reals and there exists a sequence of functions so that f_n (t)=f(nt) for n=1,2,3... also the sequence of functions is equicontinuous on [0,1].

    what does show about f? I show f is constant on the nonnegative reals

    Let epsilon (from here onwards we shall call it e)>0, then because f_n is equicontinuous there exists a delta>0 s.t. d(x,y)<delta implies |f_n(x)-f_n(y)|<e for all n as long as x,y in [0,1]. Now fix any two positive numbers x,y. since x/n and y/n go to zero there exists a N so that |x/n-y/n|<delta for all n>N and that they are both in [0,1]. Then this means that |f_n(x/n)-f_n(y/n)|<e for all n>N. This means |f(x)-f(y)|<e. Since epsilon is arbitrary we see that f(x)=f(y).

    I feel like there should be more to this or maybe that I'm wrong altogether.

    What examples satisfy the hypothesis?
     
    Last edited: Nov 29, 2007
  2. jcsd
  3. Nov 29, 2007 #2
    f must be constant on [tex][0,\infty)[/tex]. The way I proved it was by looking at f(0) = f_n(0). Fix any c in [tex][0,\infty)[/tex], and e > 0. Find d > 0 such that |x-y|<d implies |f_n(x)-f_n(y)| < e for all n, x,y in [0,1]. Then choose n such that |c/n| < min{d,1}. Then |f(0)-f(c)| = |f_n(0)-f_n(c/n)| < e. It follows that for any c in [tex][0,\infty)[/tex], and any e > 0, |f(0)-f(c)| < e. Hence f is constant on [tex][0,\infty)[/tex].

    (edit: changed the proof to restrict c to [tex][0,\infty)[/tex]).
     
    Last edited: Nov 29, 2007
  4. Nov 29, 2007 #3
    Actually, I didn't notice the restriction "equicontinuous on [0,1]"... when I did this problem I must have glanced over that.. I assumed f_n was equicontinuous on R.

    So all I really proved was that f is constant on [tex][0,\infty)[/tex].
     
  5. Nov 29, 2007 #4
    correction

    And after a second look, the condition that f is constant on [tex][0,\infty)[/tex] is necessary and sufficient to satisfy the hypothesis.

    That is, you can show that as long as f is constant on [tex][0,\infty)[/tex], regardless of what f does on [tex](-\infty,0)[/tex] (even not continuous), then [tex]f_n(x)=f(nx)[/tex] will be equicontinuous on [0,1]. This is actually easy to see: [tex]f_n(x)=f(nx)=C[/tex] for all x in [0,1], hence it's obviously equicontinuous there.
     
    Last edited: Nov 29, 2007
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