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Prove: Aka, need help

  • Thread starter Math_Geek
  • Start date
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1. Homework Statement
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. Homework 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.
 

Answers and Replies

morphism
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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.
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.
 
23
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so show since f is continuous at 0, there exists an epsilion >0 such that and deal with the two functions?
 
morphism
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There's just one function here: f. You want to prove that it's continuous (on all of R).
 
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ok so how does the f(x1)+f(x2) play into it?
 
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].
 
tiny-tim
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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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