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Math Help - Continuity problem 2

  1. #1
    Junior Member shannu82's Avatar
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    Exclamation Continuity problem 2

    Suppose that the function f and g are discontinuous at a. Which of the following are true...
    i) f+g can never be continuous at a.
    ii) 2f can never be continuous at a.
    iii) f/g can never be continuous at a.
    If it is true give a valid proof, if its false then give a counter example.

    Any help would be appreciated. I believe i and ii is true but iii is false but don't quite understand how to show the proof...
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  2. #2
    Senior Member
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    Arrow

    If
    f=0 x<0
    f=1 x>=0

    g=1 x<0
    g=0 x>=0

    f+g=1 everywhere.
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  3. #3
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by shannu82 View Post
    Suppose that the function f and g are discontinuous at a. Which of the following are true...
    i) f+g can never be continuous at a.
    ii) 2f can never be continuous at a.
    iii) f/g can never be continuous at a.
    If it is true give a valid proof, if its false then give a counter example.

    Any help would be appreciated. I believe i and ii is true but iii is false but don't quite understand how to show the proof...
    (i) and (iii) are false. zzzoak gave you a counter example for (i). try to find one for (iii) -- Hint: i'd let f = g = ..., that is, make f and g the same function. Play around with that. To prove (ii) is true, go back to the definition of what it means to be continuous: \displaystyle \lim_{x \to a} f(x) = f(a), provided \displaystyle f(a) is defined and all that carry on... personally, i'd use a proof by contradiction, or contrapositive, if that tickles your fancy.
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