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Analysis calculus proof kick start question

  • Thread starter gaborfk
  • Start date
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1. Homework Statement
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. Homework 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:

Answers and Replies

229
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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
 
53
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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
 
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].
 
tiny-tim
Science Advisor
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Hi gaborfk! :smile:

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

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