1. Not finding help here? Sign up for a free 30min 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!

Convergence and continuity question

  1. Nov 21, 2006 #1
    Can anyone help with this question?

    Let (f_n) be a sequence of continous functions on D a subset of R^p to R^q s.t. (f_n) converges uniformly to f on D, and let (x_n) be a sequence of elements in D which converges to x in D. Does it follow that (f_n(x_n)) converges to f(x)?

    My proof goes like this:
    Since (f_n) are all continous and uniformly converge on D to f we know that f is continous on D. By a thm. So f is continous at all the values on (x_n) and is continous at x. Since (f_n) is uniformly convergent to f there exist a natural number k(e) s.t. for all e>0 n>=k(e) x an element of D that ||f_n(x)-f(x)|| < e but since (x_k) and x are both elements of D we know ||f_n(x_k)-f(x_k)||<e for all natural numbers k. So we know f_n(x_k) converges to f(x_k) (By a thm that says if f is continous at a then if (x_n) is any sequences in D(f) that converges to a,k the the sequence (f(x_n)) converges to f(a)) We know that (x_n) an element of D is a sequence which converges to x Thus f(x_n) converges to f(x). Thus f_n(x_n) converges to f(x_n) which converges to f(x). Thus (f_n(x_n)) converges to f(x).

    My problem is I can see that if (f_n(x)) converges to f(x) and if f(x_n) converges to f(x) then the result seems clear that (f_n(x_n)) converges to f(x) but in my proof it doesn't seem clear to just say that. Does this make sense or is there a more precise way to say it?

    thanks
     
  2. jcsd
  3. Nov 21, 2006 #2

    StatusX

    User Avatar
    Homework Helper

    If you want to be rigorous you should use the epsilon delta definition. You can get |f(x)-f_n(x_n)|=|f(x)-f(x_n)+f(x_n)-f_n(x_n)| <=|f(x)-f(x_n)|+|f(x_n)-f_n(x_n)| by the triangle inequality. Can you see what to do from here?
     
  4. Nov 22, 2006 #3
    I got it now. thanks
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Convergence and continuity question
  1. Continuity question (Replies: 7)

Loading...