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Motive behind the proof in f(x+y)=f(x)+f(y) is continuous

  1. Oct 11, 2011 #1
    1. The problem statement, all variables and given/known data
    The problem is: Suppose f is a function with the property f(x+y)=f(x)+f(y) for x,y in the reals. suppose f is continuous at 0. show f is continuous everywhere.

    I saw this post as an alternative solution that doesn't use epsilon-delta:
    https://www.physicsforums.com/showpost.php?p=1987108&postcount=4

    But I'm not sure how one would be motivated to set h=x-a...
    I know it's a nice step and all that ultimately proves the statement, but what would be the main motivation for this?

    NOTE: I understand this solution fully. I just want to know how one is motivated to think this way...
     
  2. jcsd
  3. Oct 11, 2011 #2

    Dick

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    The best answer is try to prove it by yourself. Forget the proof you just saw. Start by trying to prove limit f(x) as x->0 must be 0. Then try and prove it your own way.
     
    Last edited: Oct 11, 2011
  4. Oct 12, 2011 #3

    HallsofIvy

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    We were given that f is continuous at x= 0 and want to show that it is continuous at x= a. So I would think the "translation" x-> x- a that maps a to 0 would be obvious.
     
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