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Analysis calculus proof kick start question

  1. Mar 24, 2008 #1
    1. The problem statement, all variables and given/known data
    Prove: If [tex]f[/tex] is defined on [tex]\mathbb{R}[/tex] and continuous at [tex]x=0[/tex], and if [tex]f(x_{1}+x_{2})=f(x_{1})+f(x_{2})[/tex] [tex]\forall x_{1},x_{2} \in\mathbb{R}[/tex], then [tex]f[/tex] is continous at all [tex]x\in\mathbb{R}[/tex].


    2. Relevant equations

    None

    3. The attempt at a solution

    Need a pointer to get started. Cannot wrap my head around it. I understand that I need to prove that the sum of two continuous functions is continous also.
     
    Last edited: Mar 24, 2008
  2. jcsd
  3. Mar 24, 2008 #2
    there is only one function here, f , it has the property that f(x + y) = f(x) + f(y) for all x, y


    hint, show f(0) = 0
     
  4. Mar 24, 2008 #3
    I know that the function is continuous at x=0. So how does showing it is continuous at zero help with showing the function with the property [tex]f(x_{1}+x_{2})=f(x_{1})+f(x_{2})[/tex] is continuous?

    Thank you
     
  5. Mar 25, 2008 #4

    HallsofIvy

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    f is continuous at x= a if and only if
    [tex]\lim_{x\rightarrow a}f(x)= f(a)[/itex].

    If h= x- a, then x= a+ h and h goes to 0 as x goes to a: that becomes
    [itex]\lim_{h\rightarrow 0}f(x+a)= f(a)[/itex].
     
  6. Mar 25, 2008 #5

    tiny-tim

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    Hi gaborfk! :smile:

    Hint: f(a + epsilon) = f(a) + f(epsilon) :smile:
     
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