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Math Help - Real Analysis Proof help dealing with continuity

  1. #1
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    Real Analysis Proof help dealing with continuity

    If R->R satisfies f(x+y)=f(x)+f(y) for all x,y in R and 0 is in C(f), then f is continuous.


    show me a great proof...

    thanks in advance
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by antoniowilliams46 View Post
    If R->R satisfies f(x+y)=f(x)+f(y) for all x,y in R and 0 is in C(f), then f is continuous.


    show me a great proof...

    thanks in advance
    A great one huh?

    This is obvious since f(0)=f(0)+f(0)\implies f(0)=0 and thus 0=f(0)=f(x-x)=f(x)+f(-x)\implies f(-x)=-f(x). Thus, |f(x)-f(y)|=|f(x-y)| now there exists some \delta>0 such that |z|<\delta\implies |f(z)|<\varepsilon. Make |x-y|<\delta
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    thats all that i need???? nothing more???
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  4. #4
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by antoniowilliams46 View Post
    thats all that i need???? nothing more???
    You need to say more but that is the entire idea.
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  5. #5
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    still need help on what to say in the proof
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  6. #6
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    i still need a proof for this one
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  7. #7
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    Do you understand Drexel proof? If you tell me where you get lost, I may be able to help you.
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  8. #8
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    I dont know where to go after abs val of x-y is less than delta.
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  9. #9
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    Let z=x-y. Then |z|=|x-y|<\delta\implies |f(z)| = |f(x-y)| = |f(x)-f(y)|<\varepsilon.

    Another way to see it is: f is continuous at x iff f(x+h)=f(x)+o(1) as h\to0. So in this case, f(x+h) = f(x) + f(h) by linearity and f(h)=o(1) by continuity at 0.
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