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Prove: Aka, need help

  1. Mar 22, 2008 #1
    1. The problem statement, all variables and given/known data
    Prove: If f is defined on the reals and continuous at x=0, and if f(x1+x2)=f(x1)+f(x2) for all x1,x2 in the reals, then f is continuous at all x in the reals.

    2. Relevant equations

    Using defn of limits and continuity

    3. The attempt at a solution
    is this like proving that the sum of two functions is continuous? I am a bit confused, this is the last on of the homework, then I can enjoy my easter.
     
  2. jcsd
  3. Mar 22, 2008 #2

    morphism

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    No. f is a function that satisfies the functional equation f(x_1 + x_2) = f(x_1) + f(x_2). You're told that f is continuous at 0, and you're supposed to use this to conclude that f is continuous everywhere.
     
  4. Mar 22, 2008 #3
    so show since f is continuous at 0, there exists an epsilion >0 such that and deal with the two functions?
     
  5. Mar 22, 2008 #4

    morphism

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    There's just one function here: f. You want to prove that it's continuous (on all of R).
     
  6. Mar 23, 2008 #5
    ok so how does the f(x1)+f(x2) play into it?
     
  7. Mar 23, 2008 #6

    morphism

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    You also know something else. f is continuous at 0.

    Now use the definition of continuity and these two facts. You might also find it helpful to prove that f(-x) = -f(x) [hint: f(0)=0].
     
  8. Mar 23, 2008 #7

    tiny-tim

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    Hint: what is f(x + epsilon) - f(x)? :smile:
     
  9. Mar 23, 2008 #8

    HallsofIvy

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    [tex]\lim_{x\rightarrow a} f(x)= \lim_{h\rightarrow 0}f(a+ h)[/tex]
    where h= x- a. Then use the fact that f(a+ h)= f(a)+ f(h).
     
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