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Regulated functions

  1. Jan 21, 2010 #1


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    1 Show that the product of two regulated functions is regulated.

    2. A function is regulated if it is the limit of a sequence of step functions.

    3. I let f,g be regulated and let a_n, b_n tend to f, g respectivley. I can show that for any x, a_n (x) . b_n (x) tends to f(x).g(x) (i.e. pointwise convergence). Is this sufficient or do I need to show uniform convergence? If so how do I go about it?
  2. jcsd
  3. Jan 21, 2010 #2
    You do need to show uniform convergence, since to show [tex]fg[/tex] is a regulated function, you need to exhibit a sequence of step functions converging uniformly to [tex]fg[/tex].

    I think one needs some additional information; is the domain a compact closed interval? The function [tex]f(x) = x[/tex], for instance, is not a uniform limit of step functions when you take the domain to be all of [tex]\mathbb{R}[/tex].

    Assuming the domain is bounded, think about [tex]\sup_x |f(x)|[/tex] or [tex]\sup_x |g(x)|[/tex].
  4. Jan 22, 2010 #3


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    Sorry I probably should have said, the two functions f,g are on the closed real interval [a,b] for some real a,b.
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