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Proof of Continuity

  1. Mar 7, 2008 #1
    I am having trouble with the following proofs. If someone can help I would appreciate it.

    Problem Statement

    Given that f, g are continuous at z, prove that

    a- f+g is continuous at z
    b- For any complex [tex]\alpha, [/tex][tex]\alpha[/tex]f is continuous at z

    There are other parts to this but if I think if I can get help on a) I think the rest will follow


    Solution


    The definition of continuous function is
    (lim x->c) f(x) = f(c)

    I have to determine if f+g is continuous at z. To do this I am proceeding as follows:

    1. (lim x->z) (f+g) = (lim x->z) f(x) + (lim x->z) g(x)
    2) Since (lim x->z) f(x) = continuous (given) and g(x) is coninuous (given) the sum will also be continuous.

    Is this mathematically sufficient or am I missing a step.
    My problem with proofs is that I use the result in my explanation which is a no-no.

    Any help will be appreciated.

    Thanks

    Asif
     
  2. jcsd
  3. Mar 7, 2008 #2
    YIOu can either use directly epsylon, delta definition of continuity of a function, or you can approach it this way:
    since f, g continuous at z, we have
    [tex]\lim_{x\rightarrow\ z}f(x)=f(z) \ \ ,\ \lim_{x\rightarrow\ z} g(x)=g(z) [/tex] now

    [tex]\lim_{x\rightarrow\ z}(f(x)+g(x))= \lim_{x\rightarrow\ z}f(x)+\lim_{x\rightarrow\ z} g(x)=f(a)+g(a)[/tex] which actually is what we want to show!!!
     
  4. Mar 7, 2008 #3

    HallsofIvy

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    So you can use that and are not required to go back to "[itex]\epsilon, \delta[/itex] to prove it?

    Isn't that simply a restatement of the theorem you want to prove? No, not exactly- I just reread it and while you just say "g(x) is continuous", you say "(limx->z)f(x)= continuous" but I can't make heads or tails of that. Surely the limit is not the word "continuous"! Perhaps you mean to say, "Since f is continuous at z, [itex]\lim{x\rightarrow z} f(x)= f(z)[/itex] and since g is continuous at z, ...", using that to show that [itex]\lim{x\rightarrow z} f(x)+ g(x)= f(z)+ g(z)[/itex] and then using that to show that f+ g is continuous. If your teacher is a real hard nose, you might need to show what "f(x)+ g(x)" has to do with "f+ g"!

    It certainly is! I would strike you across the nose with a rolled up newspaper!

     
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