1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Is C^l closed in C^0?

  1. Apr 12, 2012 #1
    1. The problem statement, all variables and given/known data

    Is [itex]C^{k}[a,b][/itex] closed in [itex]C^{0}[a,b]?[/itex]

    3. The attempt at a solution

    [itex]C^{k}[a,b][/itex] is obviously a subset of [itex]C^{0}[a,b][/itex].

    My gut feeling says no. I thought the best way would be to construct a function [itex]f_{n}(x)[/itex] which converges to [itex]f(x)[/itex] and where [itex]f_{n}(x)[/itex] is in [itex]C^{k}[a,b][/itex] but [itex]f(x)[/itex] is not.

    I thought maybe [itex]f_{n}(x)=x^{k+1}sin(\frac{1}{nx})[/itex] would do it since it's not k+1 differentiable at 0. But then [itex]f(x)=0[/itex] which can be differentiated infinitely (since each derivative is 0).
  2. jcsd
  3. Apr 12, 2012 #2
    You are definitely on right track. Try writing your sequence as a series
    [itex] f_n = \sum_{i=0}^n a_n \sin(b_n x) [/itex] and then choose an and bn so that limit of fn exists but f'n diverges.

    ... Or just google "Weierstrass function", if you're lazy :)
  4. Apr 12, 2012 #3
    If you're asking such questions, then you should always say which metric you're working with.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook