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Proof of continuity

  1. Mar 30, 2012 #1
    1. The problem statement, all variables and given/known data
    If the function f+g:ℝ→ℝ is continuous, then the functions f:ℝ→ℝ and g:ℝ→ℝ also are continuous.


    2. Relevant equations



    3. The attempt at a solution
    Ok, just learning my proofs here, so I'm not sure if my solution is cheating or not rigorous enough. take f(x)= {-1 if x≥0, 1 if x<0} and take g(x)= {1 if x≥0, -1 if x<0}. Then the function (f+g)(x) is a constant function equal to 0 everywhere. since g(x) and f(x) are both discontinuous at x=0, this is a contradiction to the given statement. Basically, I dont know if it is ok to use a piecewise function like this to disprove a statement.
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Mar 30, 2012 #2

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Yes. That's perfect. You've found discontinuous functions f and g such that f+g is continuous. So the statement is false.
     
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